COURSE 002: INTRODUCTORY MATHEMATICAL ECONOMICS

DELHI SCHOOL OF ECONOMICS

HANDOUT ON PRILIMINARIES

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Welcome to D’School! These notes are intended to warm-up our incoming MA students. We assume that you have taken a course on math-eco before. We hope that most of the material in these notes will be a review of what you already know, before we take you to a higher level. If you have never taken such a course before, you are strongly encouraged to go through this material presented and/or mentioned here on your own. If some of the material is unfamiliar, do not be intimidated! We hope you find these notes helpful! If not, you can consult the references listed in the syllabus, or any other textbooks of your choice for more information or another style of presentation.

Now Cheers!

Topics covered in this handout -

Elementary Logic Set theory Functions, types, Concavity, Convexity, Contours Group theory, Ring, Field

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1.1.Elements of logic 1.1.1. Sentential Logic

Assertions or statements have the property of being either true or false. In sentential logic we consider only those sentences which are assertions. Examples of Sentences:

(i) Sun rises in the east. (ii) Two and two make five. (iii) What time is it? (iv) One ought not to tell lies.

The !rst two are statements and the last two are not.

1.1.2. Logical Connectives (i) Negation (!!): Negation of a true statement is false and that of a false one is true.

Schematically, A !!A T F

F T (ii) Conjunction (""## $$ ##%%&&''((##))): A conjunction of two statements is true if and only if

both the constituent statements are true. Schematically, A B (A ! B) T T T T F F F T F F F F

The word ‘but’ has about the same sense as ‘and’.

(iii) Disjunction (**): In common usage ‘or’ is used both in exclusive and inclusive senses. In logic ‘or’ is always used in the inclusive sense. A disjunction of two statements is false if and only if both of its constituents are false. Schematically,

A B (A ** B) T T T T F T F T T F F F

(iv) Conditional or Implication (!): A Conditional is a statement of the form A " B (read as ‘If A then B’) where A and B are statements. A is called the antecedent and B the consequent. A conditional is false if and only if the antecedent is true and the consequent is false. Schematically,

A B (A ! B) T T T

T F F F T T F F T

Again, consider P " Q. If the statement P is true, then the statement Q is true. But if P is false, Q may be true or false. For example, on real numbers, let P = x > 0 and Q = x2 > 0. If P is true, so is Q, but Q may be true even if P is not, e.g. x = - 2. We club these together and say P " Q.

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(v) Biconditional ( " ): A # B (‘A if and only if B’) is true if and only if A and B

have the same truth-value. Schematically, A B (A " B) T T T T F F F T F F F T

A classical example of a non-truth-functional connective is that of possibility. For example, the sentence:

(1) It is possible that there is life on Mars

is true under any liberal interpretation of the notion of possibility; but then so is the sentence:

(2) It is possible that there is not any life on Mars.

On the other hand, the sentence:

(3) It is possible that 2+2=5

is ordinarily regarded as false.

1.1.3. Necessity and Sufficiency

- Necessity: A is necessary for B , !A must hold or be true in order for B to hold good

or to be true, !B is true requires that A must also be true! “A if B” or “A is implied

by B” (A!B).

If A is not true, B must be not true. But that doesn’t mean

that if B is not true, A must be not true.

! A is not true ! B must be not true ! B is not true is necessary for A is not true.

i.e. ~A ! ~B.

- Sufficiency: “A is sufficient for B” means that if A holds, B must hold. We can say “A

is true only if B is true”; or A implies B (A!B).

B is not true ! A must be not true. ! B is not true is sufficient for A is not true i.e. ~B ! ~A.

- Both Necessity and Sufficiency: A"B

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1.1.4. Theorems and Proofs

“A ! B” : A is true! B must be true

- Here, A is called “premise” and B is call “conclusion”

- Constructive proof / Direct proof: Assume that A is true, deduce various

consequences of that, and use them to show that B must hold.

- Contrapositive proof: Assume that B does not hold, then show that A cannot hold.

~B ! ~A is the contrapositive of A ! B.

1.1.5. Theorems, Assumptions and Conclusion

- A Theorem: is simply a statement deduced from other statements and should be a

concept familiar from courses in mathematics.

- Theorems provide a compact and precise format for presenting the assumptions and

important conclusions of sometimes lengthy arguments, and so help to identify

immediately the scope and limitations of the result presented.

1.1.6. Tautology: A tautology is a statement that is true for all logically possible

combinations of truth-values of the constituent statements.

Let us call a sentence atomic if it contains no sentential connectives.

We can also define the notion of tautology as follows: A sentence is a tautology if and only if the result of replacing any of its component atomic sentences (in all occurrences) by other atomic sentences is always a true sentence.

Tautological Implication: A tautologically implies B if and only if (A " B) is a tautology.

Tautological Equivalence: A is tautologically equivalent to B if and only if (A # B) is a tautology.

Examples of tautologies:

(i) A " A

(ii) A " (A * B)

(iii) (A " B) # (B " A)

(iv) (A * B) # (B * A)

(v) (A " (B " C)) # ((A " B) " C)

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(vi) (A " (B * C)) # ((A " B) * (A " C))

(vii) (A * (B " C)) # ((A * B) " (A * C))

(viii) (!(A " B)) # (!A *!B)

(ix) (!(A * B)) # (!A "!B)

(x) (A " B) # (!B "!A)

(xi) (A # B) # ((A " B) " (B " A))

(xii) (A " C) " (B " C) " ((A * B) " C)

(xiii) (A " B) # [!(!A *!B)]

(xiv) (A " B) # [!(A "!B)]

(xv) (A * B) # [!(!A "!B)] # (!A " B)

(xvi) (A * B) # (!A " B)

(xvii) (A " B) # [!(A "!B)]

(xviii) (A " B) # (!A * B)

(xix) (A # B) # [!(A "!B) "!(B "!A)]

(xx) (A # B) #! [! (!A * B) *! (!B * A)]

(xxi) (A # B) #![(A " B) "!(B " A)].

Inference: Q logically follows from P if P ! Q is a tautology, or iff the conditional P ! Q is universally valid. In other words:

Q logically follows from P if Q is tautologically implied by P.

The consistency of a set of premises whose logical structure may be expressed by sentential connectives alone may be determined by a mechanical truth table test. The truth table for the conjunction of the premises is constructed. If every entry in the !nal column is ‘F’ then the premises are inconsistent. If at least one entry is ‘T’ the premises are consistent.

Quanti#ers: x " 5 is not a statement. In order to change x " 5 into a statement one has to replace the variable by some constant or use a quanti!er. Consider the following two statements:

(i) For any number x, if x is divisible by 2 and 3 then x is divisible by 6. [In symbols: (! numbers x) (x is divisible by 2 and 3 " x is divisible by 6)].

(ii) For any number x, there exists a number y such that y " x. [In symbols: (! numbers x) (+ a number y) (y " x)].

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In statement (i) the variable x is bound by a universal quanti!er and in statement (ii) the variable x is bound by a universal quanti!er while y is bound by an existential quanti!er.

Example: Let S be set of natural numbers. Then, !(, x " S) (x is divisible by 5) # (#$x$" S) (x is not divisible by 5). %(#$x$" S) (x is divisible by 5) # (,#x - S) (x is not divisible by 5).

1.2. Elements of Set Theory 1.2.1. Notation and Basic Concepts

- Set: A set is a collection of objects/ elements. Elements may be numbers or vectors.

Examples: S = {1, 7, 9, 4}. A = {1, red colour}. N = {x | x is a natural number}.

The members of a set are called its elements. If x is an element of set A we say that x is in A and write x " A.

- Subset: A is a subset of B (i.e. A & B) if and only if for every x, if x " A then x " B.

- Proper Subset: Set A is a subset of set B if every element of set A is also an element

of set B. Notation: A! B.

- Empty set: S is an empty set if it contains no elements at all. Notation: S = !

- Complement Set: The complement of set S in a Universal set U is the set of all

elements in U which are not in S. Notation: Complement set of S: S# OR cS OR S c.

- Union: Notation { | }S T x x S or x T! " # # . In general: i I iS!" (I is Index set).

- Intersection: Notation { | }S T x x S and x T! " # # . Or i I iS!! (I is Index set).

- Index Set: The set of interger number starting with 1. S = (1, 2, 3, 4..., n). Notation:

...}3,2,1{!I

- Difference of Sets: A – B = { | and }x x A x B! " . Also called the complement of B relative to A. More familiar is the notion of the universal set ! , and cB B!" = .

- Set Equality: A is equal or identical to B if and only if A and B have the same elements. A = B $ [(A & B) ' (B & A)].

- n – Space: is the Set Product : 1 2... {( , ,..., ) | }nn iR R R R x x x x R= ! ! ! " # i =1,2...,n.

- Non-Negative Orthant: 1 2{( , ,..., ) | 0}n nn iR x x x x R+ ! " # "

- The Product of two sets OR, Cartesian Product: Cartesian product of two sets A and B is defined as the set A ( B = {(a, b) | a " A ' b " B}. Or, },|),{( TtSstsTS !!"#

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Example: Suppose A = {blue colour, red colour} and B = {7, 5, 2}. Then, A ( B = {(blue colour, 7), (blue colour, 5), (blue colour, 2), (red colour, 7), (red colour, 5), (red colour, 2)}.

1.2.2. Some Important Set Identities and Other Principles

(1) A . ! = A Note here: ! is null set, and ! is universal set.

(2) A $ ! = A

(3) A . B = B . A

(4) A $ B = B $ A

(5) A . (B $ C) = (A . B) $ (A . C)

(6) A $ (B . C) = (A $ B) . (A $ C)

Proof. of (6) Suppose x !LHS. So, x ! A and x ! (B $ C). So x ! A and either x ! B or x ! C or both. So, either x ! (A $ B) or x ! (A $ C) (or both). Converse?

(7) A .!A = !

(8) A $!A =!

(9) A . A = A

(10) A $ A = A

(11) A . ! = !

(12) A $ ! = !

(13) ! ! !

(14) !!A = A

(15) A = !B " B = !A

(16) A . B ! ! " A! ! * B! !

(17) A $ B ! ! " A! !

(18) A . (B . C) = (A . B) . C

(19) A $ (B $ C) = (A $ B) $ C

(20) A . (A $ B) = A

(21) A $ (A . B) = A

(22) !A ! A

- Distributive Laws "

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(23) !(A . B)= !A $!B

(24) !(A $ B)= !A .!B

(25) A $ !A =!

(26) A ! (A $ B)= A ! B

(27) A $ (A ! B)= A ! B

(28) (A ! B) ! B = A ! B

(29) (A ! B) ! A =!

(30) (A ! B) . B = A . B

(31) (A . B) ! B = A ! B

1.2.3. Relations and Functions

Consider the two sets: set S (s1, s2,...) and set T (t1, t2,...).

The product of the two sets: S ! T = {(s,t)| s ! S and t ! T} , called Order pairs.

Any Collection or Set of Order pairs is said to constitute a Binary Relation (S/T) of the

sets S and T. Binary Relation: A binary relation from set S to set T is a subset of S( T.

Meaningful Relationship: (sRt) : The Set of the order pairs that are constituted by elements

of the sets S and T are meaningful.

},|),{( TSsRtandTtSstssRt !"##$

Example: },|),{(2 SySxyxSSS !!==" " " {( , ) | , , }x y x S y S x y! " = # # "

Function: A relation f from S to T , notationally f : S ! T , is called a function iff for every x

" S, there is a unique y " T such that (x, y) " f.

In other words, a relation from S to T is a function iff –

[(! x " S)[(#$y " T )((x, y) " f)] ' (! x " S)(!$y, y’" T )[(x, y) " f ' (x, y’) " f ! y = y’]].

Example: Consider sets A = {2,3,4,9} and B = {1,7,6,4,8}.

A ( B = {(2, 1), (2, 7), (2, 6), (2, 4), (2, 8), (3, 1), (3, 7), (3, 6), (3, 4), (3, 8), (4, 1), (4, 7), (4, 6), (4, 4), (4, 8), (9, 1), (9, 7), (9, 6), (9, 4), (9, 8)}.

Let: R1 = {(2, 1), (3, 6), (9, 4)}. R2 = {(2,4), (2,6), (3,1), (4,7), (9,8)}.

R3 = {(2, 8), (3, 1), (4, 4), (9, 7)}. R4 = {(2, 8), (3, 1), (4, 4), (9, 4)}.

Among these four relations only R3 and R4 are functions. Why? Check yourself.

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Image: Let f : S ! T be any function, and let A be a subset of S. De!ne f(A)= {y | for some x " A, (x, y) " f}. We call f(A) the image of A in T .

Inverse of a relation: Given any relation R from A to B, we de!ne R%1 = {(y, x) | (x, y) " R}. The relation R%1 from B to A is called the inverse of R.

1.2.4. Properties of Binary Relations: Let R be a binary relation on a set S. We de!ne:

(! x, y " S) (xPy $ xRy '$% yRx) : Preference relation

(! x, y " S) (xIy $ xRy ' yRx). : Indifference relation

• Completeness: A relation R on S is complete if, for all distinct elements x and y in S,

xRy or yRx [the order pairs (x,y) or (y,x)] are all meaningful. Ex1: x||y

• Re$exivity: R on S is re&exive i0 (! x " S)(xRx). • Symmetry: R on S is symmetric i) (! x, y " S)(xRy ! yRx). Ex: x"y • Transitive: A relation R on S is Transitive if, for all three elements x, y and z in S, xRy

and yRz implies xRz."R on S is transitive i) (! x, y, z " S)(xRy " yRz ! xRz). Ex1. • Odering: A binary relation which satisfies all properties of Completeness, Reflexivity

and Transitivity is called an Ordering.

A relation which is reflexive, transitive and symmetric is called an Equivalence relation.

Following are the some other properties of set –

- Irre$exivity: A relation R on S is reflexive if, for all elements x in S, xRx (ordered

pairs (x, x) are meaningful). R on S is irre&exive i0 (! x " S)(%xRx).

- Connected: R on S is connected i) (! x, y " S)(x = y ! xRy * yRx).

- Asymmetry: R on S is asymmetric i) (! x, y " S)(xRy !%yRx).

- Anti-symmetry: R on S is Anti-symmetric i) (! x, y " S)(xRy '$yRx ! x = y).

- Quasi-transitive: R on S is quasi-transitive i) (! x, y, z " S)(xP y " yP z ! xP z).

- Acyclic: R on S is acyclic i0 (!$n " N %{1, 2})(!$x1, x2, ... xn " S)(x1Px2 ' x2Px3 ' ... '

xn%1Pxn ! x1Rxn), where N is the set of positive integers.

2.1. Mapping and Functions: 2.1.1. Mapping

Given two non-empty sets A and B, if it be possible to associate or match each element of the set A with a single element of B by some specified manner f, then this matching procedure is called mapping or correspondence.

This mapping or map is denoted by f: A ! B.

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Consider the sets A ={1, 2, 3, 4}, B ={4, 8, 12, 16}. Here we observe that with each element of the set A, we can match element of the set B which is a number four times as large. We call this the mapping f and denote the match by writing: f: x!4x.

We define mapping (or function) in an equivalent form as below –

Let A and B be two non-empty sets. A sub-set f of A!B is called a function or mapping from A to B, if, to each a !A, exists a unique b! B such that the ordered pair (a,b) ! f. The set A is called the domain and the set B is called the co-domain of the mapping f. We see that, to each a in A, f associates an element b in B uniquely.

In general, if a !A and b !B, then we say that b is the f-image of a and write f(a) = b. a is called the pre-image of b. Note that there may be more than one pre-image of b.

A map f: A!B is well defined, if

(i) a !A ! f(a) !B, (ii) any element a !A!unique element f(a) ! B, (iii) two or more than two elements of A may have same image in B.

If A = {2, 3, 4} and B = {5, 7, 8} and if f(2) = 5, f(3) = 7, then f: x ! 2x + 1 does not define a mapping as 4!A has no image in B.

2.1.2. Different types map:

Consider the following mappings :

A = {1, 2, 3, 4, ......}, B ={ 1,1/2,1/3, ',...... }; thus f: x!1/x.

A = { 1,2,3,4, ...... }, B = {I, 4, 9, 16, ...... }; thus f: x!x2 .

In the above two mappings, one element a !A maps to one element b ! B. Such a mapping is called one-one In this case distinct element of the domain has distinct in the co-domain.

Thus a mapping f is called one-one or injective, if different elements of the set have different f-images in B. Thus, if a1, a2!A, then a1 ! a2! f(al) ! f(a2) in B, or equivalently f(a1)= f(a2) ! a1=a2.

If f be injective then each element of B has at most one image in A. The graph of f, denoted by f*, is the sub-set of A! B by {a, f(a) : a !A and f(a) ! B }.

The range of f is the set of all images under f and is denoted by f(A) or {f(a)} for all a !A. Evidently f(A)= {b:b=j(a) for some a !A and b !R}. In general, f(A)! B. Let the domain A be {a, b, c, d} and the co-domain B be {a, b, c}· Then for the function f from A ! B as given by f = {(a, b), (b, c), (c, c), (d, b)}, the range f (A) is {b, c}.

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Consider a set A = {A1, A2, A3, A4, A5}, the elements being the students of your class and a set B= {50, 55, 61}, the elements being the marks obtained by the students. It is possible that more than one student may secure the same mark; hence here more than one member of the domain may have the same image.

This type of mapping is called many-one-mapping. While matching, if every element of the co-domain be an image of some element of the domain and no element of the co-domain remains unused, then the mapping is from the domain onto the co-domain. Such a mapping is called onto mapping.

On the other hand, if at least one element in the co-domain be not an image of some element of the domain, then such a mapping is not an onto mapping and is called an into mapping. If A = {1, 2, 3, 4}, B ={2, 3, 4, 5} be two sets, then f : x - x + 1 maps A onto the set B. Again, if A be the set of all even positive integers, then f: x ! x + 1, x ! A, is not onto mapping of A to the set B of all positive integers, odd or even.

Notice that for an into mapping {f(x)}! B for all x!A, while for an onto mapping {f(x)} = B for all x!A. Also notice that the range is always a sub-set, not necessarily a proper sub-set, of the co-domain. We give diagramatic examples of different types of mapping here –

Onto mapping is also known as surjective mapping. The one-one and onto mapping is known as bijeciive mapping. In this case the two sets will contain the same number of elements.

Besides these, there are many-one onto and many-one into mappings.

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Let f : A!B and g : B !C be two maps. Then the two mappings f and g are said to be equal, if and f(x) = g(x) for all x!A.

In this case the domains of the mappings must be the same.

Let A be a non-empty set and f be a mapping such that, by f: A !A each element A is mapped on itself. Then f is called the identity mapping and f(x) = x for all x !A. If A={I, 2, 3}, then f= {(1, 1), (2, 2), (3, 3)} is an identity mapping of A. Identity mapping is always one-one onto. It is denoted by IA .

The mapping f is called constant, if the range of f be a singleton set, that is, f(a) = b for all a!A and b!B. Two sets A and B are said to be cardinally equiva1ent, if there exists a mapping f from A to B which is one-one and onto. We denote this by A~ B. These two sets are said to be equipotent with each other.

Let I be the set of positive integers and E be the set of all even integers. Consider the mapping f: I! E as given by f(x) = 2x for all x!I. Then I is equipotent with E for the mapping is one-one onto. The two sets I and E are cardinally equivalent. If a set A be cardinally equivalent to the set of natural numbers N={1, 2, 3, 4, ...... } then A is called a denumerable set. A set is said to be countable, if it be finite or denumerable.

The set A = { a2 : a !N} is denumerable for the mapping f: N ! A as given by f(a)=a2, a!N is one-one and onto.

2.1.3. Inverse mapping

Let f: A!B be a map and b!B be arbitrary. Then the inverse of the element b, which is denoted by f –1(b) is defined as a consisting of those elements of A which have b as their images. Obviously, f –1(b)!A.

Let f: A!B be a one-one onto mapping. Then the mapping f –1(b):B!A which associates, to each element b!B, the element a!A, such that f(a)=b, is called inverse mapping of the map f. Thus f –1(b)={ a!A: f(a)=b}. The map f –1(b) may be a singleton set, or a set consisting more than one element. If f be one-one onto, then f –1(b)=a ! f(a)=b.

2.2. Functions

The property of convexity is most often assumed ! analysis is mathematically tractable and

results are clear-cut and well-behaved.

Images & Preimages: Let f: A B.

Definition: Let A0 ! A. The image of A0 under f, denoted f(A0), is the set {b| b = f(a), for

some a"A0}.

Let B0 ! B. The preimage of B0 under f, f -& (B0) = {a| f(a) "B0}.

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So the image of a set is the collection of images of all its elements, and the preimage or inverse image of a set is the collection of all elements in the domain that map into this set.

Example: For the function f(x) = x', let A0 = [-2,2]. Therefore f(A0) = [0,4]. .

Let B0 = [-2, 9]. Then, f -& (B0) = [-3, 3].

Facts. Let f : A B, and A0, A1 ! A, also B0, B1 ! B. Then we claim that -

(i) B0!B1 # f -& (B0)!f -& (B1).

(ii) f -& (B0* B1) = f -& (B0) * f -& (B1), where * can be U, , -.

i.e., f -& preserves set inclusion, union, intersection and difference. f only preserves the first two of these. The 3rd holds with ! and the 4th with $. To find counterexamples, many-to-one functions (to which we now move) are helpful.

Proofs of the above claims are homework.

Injective & Surjective Functions:

Definition: f: A B is injective (one-to-one) if [f(a) = f(a )]#[a = a ]. This is

surjective (onto) if for every b " B, there exists a " A s.t. b=f(a). A function that is both of

these is called bijective.

For example, f: % % defined by f(x)=x( is many-to-one and not surjective. If the domain is

%+ instead, then f is injective, and further if the codomain is %+, then it is bijective.

Properties of Inverse of a Function:

Let f : A B, and A0!A, B0!B. Then we claim that -

(i) f -& (f (A0))$A0; equality holds if f is injective.

(ii) f (f -& (B0)) !B0; equality holds if f is surjective.

You should also show by examples that equality does not in general hold.

Now we use the f -& to mean something different, namely the inverse function.

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If f is bijective, then define a function f -&: B A by f -& (b)=a if a is the unique element of A

s.t. f(a)=b. Note that f -& is also bijective.

Indeed, suppose b b and f -& (b)= f -& (b ) = a. Then f(a)=b and f(a)=b which is not

possible. So f -& is injective. Moreover, for every a"A, there is a b"B s.t. b=f(a), since f is a

function. So, there is a b"B s.t. f -& (b)=a. So f f -& is also surjective; hence it is bijective.

2.2.1. A Function is a Relation that associates each element of one set with a single, unique

element of another set. f: D ! R; where D is the Domain and R is called the Range.

Convex set in Rn : S!Rn is a conxex set iff for all x1 , x2 ! S, we have: t. x1 + (1- t). x2 ! S for all t in the interval 10 !! t . Geometrically, if x, y - Rn, then {z - Rn : z = )x + (1 % )) y for ) - [0, 1]} constitutes the straight line connecting x and y. So a convex set is any set that contains the entire line segment between any two vectors in the set.

Theorem: Intersection of Convex Sets is Convex. Can you prove this?

Theorem: The union of two convex sets is not necessarily convex. Why not?

A vector z - Rn is a convex combination of x1, ..., xm

- Rn if

z = ! j x

j

j=1

m

" for some )1, …., )m* 0 with 1

1m

jj!

=

="

In the figure below –

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

15

Theorem 1: A set X 1 Rn is convex if and only if it contains any convex combination of any

vectors x1,...,xm- X.

Proof. The proof is by mathematical induction on m. For m =1, the only convex combination of vector x is x itself. So the basis statement for m =1 is true. The induction step is to suppose that the proposition is true for m % 1> 0 vectors, and then to show that this implies the proposition is true for m vectors. So consider any convex combination ( mj=1 )jxj of m vectors contained in X. Since m " 2 and each )j " 0 with ( mj=1 )j =1, we may suppose WLOG that )m < 1. Then since

! j

1" !m

=! jj=1

m"1#1" !m

=1" !m

1" !m

= 1j=1

m"1

# ,

the induction hypothesis implies that y +1( )1

m jjj

m

x X!!=

"#$ . Then the definition of a convex

set implies

The propoposition then follows from mathematical induction. QED.

Given any set X 1 Rn , the convex hull Co(X) is the intersection of all

convex sets that contain X. Since the intersection of any two convex

sets is convex, it follows that the convex hull is the smallest convex

set that contains X.

• If X is convex, then Co(X)=X. Why?

Observation 1: Suppose X 1 Rn . Then Co(X) is the set of all convex combinations of

vectors in X. Proof. Lemma 1 implies that any convex combination of elements x

1,..., x

m - X must be

contained in Co(X). To show that Co(X) contains only vectors that are convex combinations

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

16

of some x1,..., x

m - X, we need to show that Y + {x - R

n+: x is a convex combination of some

x1,..., x

m - X} is a convex set. So consider and y, z " Y. Then, by definition, there is a set of

vectors y1,...,ym " X and list of nonnegative numbers *1,...,*m with (*i =1 such that y = (*iyi.

Similarly, there is a set of vectors z1,...,zr" X and list of nonnegative numbers +1,...,+r with (+i

=1 such that z = 1

r iiiz!

=" . Then for any ) - [0,1],we have

)y+(1% ))z = ) 1

m

i=! ,iyi + (1% )) 1

r

i=! -izi

=1

m

i=! ),iyi + 1

r

i=! (1% ))-iz i

which, since )1

m

i=! ,i +(1% )) 1

r

i=! -i = )+(1% ))=1, implies that )y +(1% ))z is a convex

combination of y1,...,yn, z1,...,zr - X.

2.2.2. Real Valued Function

Definition of RVF: f: D!R is a real valued function if D is any set and RR ! .

We here restrict our attention to real valued functions whose domains are convex sets.

Assumption: Real Valued Function over Convex Sets.

Let f: D!R is a real valued function where D nR! is a convex set and R R! .

Increasing Functions: f: D!R is increasing function whenever f(x0)! f(x) ! xx !0 , and

xx !0 . We say f is strictly increasing function whenever f(x0) >f(x) ! x0 ! x, and xx !0 .

Note: xx !0 means that at least one of components of vector x0 is greater than the same

ordered component of vector x.

Decreasing Functions: f: D! R is decreasing function whenever f(x0)! f(x) ! xx !0 , and

xx !0 . We say f is strictly decreasing function whenever f (x0) < f(x) ! xx !0 , and xx !0 .

2.2.3. Related Sets

The graph of a function is a related set which provides an easy and intuitive way of thinking

about the function. There are some sets related to a function.

Level Sets: L(y0) is a level set of the real valued function f: D ! R iff 0 oL(y ) { | , ( ) y }x x D f x= ! = , where y0 R! . (The Contour Set)

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

17

Examples: Isoquant curve, Indiferrent curves, Isoprofit curve…

Usage of Level Sets:

- Reducing by one the number of dimesions needed to represent the function.

- Two indifferent level sets of a function can never cross

Level Sets Relative to a Point x0

L(x0) is a level sets relative to point x iff L(x0) = {x|x D! , f(x) = f(x0)}.

Superior and Inferior Sets

1. S(y0) = {x | x D! , f(x) ! y0} is call superior set for level yo.

2. I(y0) = {x | x D! , f(x) ! y0} is call inferior set for level yo.

3. S.( y0) = {x | x D! , f(x) > y0} is call strictly superior set for level yo.

4. I. (y0) = {x | x D! , f(x) < y0} is call strictly inferior set for level yo.

Theorem: Superior, Inferior, and Level Sets : For any f: D R! any Ry !0 :

1. L(y0) ! S(y0)

2. L(y0) ! I(y0)

3. L(y0) = S(y0) ! I(y0)

4. S.(y0) ! S(y0)

5. I. (y0) ! I(y0)

6. S. (y0) ! L(y0) = O

7. I. (y0) ! L(y0) = O

8. S. (y0) ! I. (y0) = O.

2.2.4. Concave and Convex functions [For the remainder of these notes, we assume

that X & Rn is a convex set. ]"

Concave Function: f : X " R is concave if for any x, z - X,we have, for all ) - (0, 1) ,

f()z +(1 % )) x) * )f (z)+(1 % )) f(x).

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

18

Strictly Concave Functions: f : X " R"is strictly concave if for any x, y " X with x = z, we"have,

for all ) - (0, 1) , f()z +(1 % )) x) > )f(z)+(1 % )) f(x).

! A!constant function is concave. Why?

! A!linear function is concave. Why?

The set of points beneath concave regions is a convex set. The set of points beneath the

non-concave region is not a convex set.

For strictly concave functions, geometrically these modifications simply require the graph

of the function to lie everywhere above the the chord connecting any two points. Rule out

any flat portions of the graph of the function.

! f : X ! R!is concave if and only if f()(z % x)+ x) " ) (f(z) % f(x)) + f(x) for all x, z "

X and ) " (0, 1) . Why?

! f : X ! R!is!concave!if!and!only!if!!f (),x + x) " ) (f(x + ,x) % f(x)) + f(x) for!all!!!x,

(x + ,x) " X and ) " (0, 1) . Why?

Theorem: Points on and below the graph of a Concave Fn always form a Convex Set

Let D nR! be a convex set and let R ! R. Let A })(,|),{( yxfDxyx !"# be the set of

points “on or below” the graph of f: D R! . Then f is concave function ! A is a

convex set.

Proof: We have to show that f(x) is a concave function implies A is convex and A is

convex implies f(x) is concave function.

First part: f(x) is a concave function ! A is a convex set. Let take any two points: (x1,y1)

and (x2,y2) in set A. Take convex combination of the two point: (xt, yt) such that:

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

19

+"

,"

xt = tx1 + (1-t) x2 and yt = ty1 + (1-t) y2 .

We need to prove that (xt, yt) is also in set A.

f(x) is a concave function ! by the definition of concave function:

f(xt) ! tf(x1) + (1-t)f(x2)

By the definition of set A, we have f(x1)! y1 and f(x2) !y2

! tf(x1) + (1-t)f(x2) ! ty1 + (1-t)y2 = yt

! f(xt) ! yt ! (xt , yt) ! A ! A is a convex set.

Second part: Proving A is convex implies f is a concave set.

Consider two any points on the graph of f(x): (x1,f(x1)) and (x2,f(x2)).

A is convex ! (xt, yt) ! A, in which :

xt = tx1 + (1-t)x2 and yt = tf(x1) + (1-t)f(x2).

We need to prove that: f(xt) ! tf(x1) + (1-t)f(x2).

According to the definition of set A:

f(xt) ! yt = t f(x1) + (1-t)f(x2). ! f(x) is a concave function.

Linear Combinations of Concave Functions

Consider a list of functions fi : X ! R for i =1,..., n, and a list of numbers *1,..., *n. The

function f +1

ni iif!

=" is called a linear combination of f1,..., fn. If each of the weights *i " 0,

then f is a nonnegative linear combination of f1,...,fn.

The next proposition establishes that any nonnegative linear combination of concave

functions is also a concave function.

Theorem 2: Suppose f1,..., fn are concave functions. Then for any *1,...,*n, for which each *i "

0, f +1

ni iif!

=" is also a concave function. If, in addition, at least one fj is strictly concave and

*j > 0, then f is strictly concave.

Proof. Consider any x, y " X and ) " (0, 1) . If each fi is concave, we have

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

20

fi [)x +(1 % )) y] " ).fi(x) + (1 % )). fi(y) !

Therefore, f()x +(1% ))y) +1

ni iif!

=" ()x +(1% ))y) " 1

nii

!=" ()fi(x)+(1% ))fi(y))

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!= )1

ni iif!

=" (x)+(1% ))1

ni iif!

=" (y)!!!- )f(x)+(1% ))f(y).

This establishes that f is concave. If some fi is strictly concave and *i > 0, then the inequality

is strict. QED.

Since a constant function is concave, Theorem 2 implies

• If f is concave, then any aff"ne transformation *f ++ with * " 0!is also concave.

• If f is strictly concave, then any aff"ne transfn *f ++ with *> 0!is also strictly!concave.

2.2.5. Quasiconcavity

Quasiconcave functions: f : X ! R is quasi-concave if for any x, z " X, we have

f()z +(1% ))x) " min{f (x),f(z)} for all ) " (0, 1).

Strictly Quasiconcave fn : f is strictly quasi-concave if for any x, z " X and x =z, we have

f()z +(1% ))x)> min{f (x),f(z)} for all ) " (0, 1). !

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#$%&"'()*(%+,!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!-).!/$%&"'()*(%+,!

- A Strictly Concave Function forbid the convex combination of two points in the same

level set also lies in that level set

Theorem 3:"Concavity Implies Quasiconcavity (Concavity ! Quasiconcavity)

A concave function is always quasiconcave. A strictly concave function is always strictly quasiconcave.

Proof. The theorem follows immediately from the observation that if f is quasi-concave, then

+-"

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+-"

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21

for all x, z " X, we have

f()z +(1% ))x)" )f(z)+(1% ))f(x) " min{f (x),f(z)} for all ) " (0, 1).

If f is strictly concave, we have for all x, y " X and x =y,

f()z +(1% ))x)>)f(z)+(1% ))f(x) " min{f (x),f(z)} for all ) " (0, 1). QED.

! If X * R,"then f : X ! R"is quasi-concave if and only if it is either monotonic or

f3rst non*decreasing and"then"non*increasing. Why?

Geometrically: When f(x) is an increasing function, it will be quasiconcave whenever

the level set relative to any convex combination

of two points, L(xt) is always on or above

the lowest of the level sets L(x1) and L(x2).

When f(x) is a decreasing function, it will be quasiconcave whenever the level set relative to

any convex combination of two points, L(xt) is

always on or below the highest of the level sets

L(x1), L(x2).

Theorem: Quasiconcave and the Superior Sets

f: D!R is a quasiconcave function if and only if S(x) i.e. the superior set relative to point

x: {x| x D! , f(x) ! y} is a convex set for all x D! .

Proof: We have to prove the both necessary and sufficient terms.

Sufficiency: Prove that f(x) is a quasiconcave function ! S(x) is a convex set.

Consider any two points x1 and x2 in set S(x). We need to prove that xt made by the convex

combination of the two vectors x1 and x2 is also in S(x) ! f(xt) ! f(x).

! according to the definition of S(x), we have: f(x1) ! f(x) and f(x2) ! f(x).

By the definition of a quasiconcave function: f(xt) ! min[f(x1),f(x2)] ! f(x)

! f(xt) ! f(x).

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22

Necessity: Prove that if S(x) is a convex set ! f(x) is a quasiconcave function.

We need to prove that f(xt) ! min[f(x1),f(x2)] under the condition that S(x) is convex. Consider

any two points x1 and x2 in S(x). Without loss of generality, assume we have: f(x1) ! f(x2).

According to the definition of S(x), we have: f(x1) ! f(x2) ! f(x).

Because f(x1) ! f(x2) ! S(x2) ! S(x1) ! x1 and x2 are both in S(x2)

! xt = tx1 + (1-t)x2 (for all t ]1,0[! ) is also in S(x2) ! f(xt) ! f(x1) ! f(x2).

Because t ]1,0[! ! f(xt) ! min[f(x1), f(x2)] ! f(x) is a quasiconcave function.

Our next theorem states that any monotone nondecreasing transformation of a quasi-concave

function is quasi-concave.

Theorem 4: Suppose f : X ! R is quasi-concave and . : f(X) ! R is nondecreasing. Then . /

f : X ! R is quasi-concave. If f is strictly quasi-concave and / is strictly increasing, then . / f

is strictly quasi-concave.

Proof. Consider any x, y " X. If f is quasi-concave, then f ()x +(1 % )) y) " min {f(x),f(y)} .

Therefore, / nondecreasing implies

.(f ()x +(1 % )) y)) " .(min {f(x),f(y)})= min {.(f(x)),.(f(y))} .

If f is strictly quasi-concave, then for x =y, we have f()x +(1 % )) y) > min {f(x),f(y)}.

Therefore, if / is strictly increasing, we have

.(f ()x +(1 % )) y)) >.(min {f(x),f(y)})= min {.(f(x)),.(f(y))} .

!Recall that for any x - X, P (x) + {z - X : f(z) * f(x)} is called the better set of x.

+."

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(&)$&*+,-."/%-)#,0-"

+-"

:;+<"

6;+<"

+."

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1-)$&*+,-."/%-)#,0-"

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23

Theorem 5: A function f : X ! R is quasi-concave if and only if P (x) is a convex set for each x " X.

Proof. (only if) Suppose f is quasi-concave. Choose an arbitrary x0 " X. To show that P (x

0) is

convex, consider any x, y " P (x0). Then, f(x), f(y) " f(x

0) and the quasi-concavity of f imply

that f()x +(1 % )) y) " min {f(x),f(y)} " f(x 0)

which implies that )x +(1 % )) y - P (x0) for any ) - (0, 1) .

(if) Suppose P (x0) is convex each x

0 " X. Now consider any x, y " X. WLOG, suppose that

f(x) 0 f(y). Then letting x = x0, we have that x, y " P (x0) and therefore )y +(1 % )) x " P (x

0)

for any ) " (0, 1) . It then follows from the definition of P (x0) that

f()y +(1 % )) x) " f(x)= min {f(x),f(y)} . QED. !

Theorem 5 is illustrated below for an increasing function f : R+2 ! R. Notice that all convex

combinations of vectors in P (x1) are also elements of P (x

1). Also notice that if f is strictly

concave, then the level set can contain no straight line segments.

Corollary 1: Suppose f : X ! R attains a maximum on X.

(a) If f is quasi-concave, then the set of maximizers is convex.

(b)If f is strictly quasi-concave, then the maximizer of f is unique.

Proof. (a) Let x0 be a maximizer of f. Then f(x) 0 f(x

0) for all x " X implies that P (x

0) is the

set of maximizers of f. If f is quasi-concave, then Theorem 5 implies that P (x0) is convex.

(b)If f is strictly quasi-concave, suppose x, y are both maximizers of f. Then x 1 y implies

1 1( )2 2

f x y+ > min {f(x),f(y)} = f(x) which implies that x is not a maximizer of f.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

24

Example: Let I > 0 be the income of some household and let p =(p1,..., pn) " nR+

denote

the vector of prices of the n goods. Then

B(p, I) - {x " nR+

: px 0 I }

defines its budget set – the set of all possible nonnegative bundles of goods it may

purchase within its budget. You can verify that y, z " B(p, I) implies )y +(1 % )) z "

B(p, I) for all ) " [0, 1]. Therefore, B(p, I) is a convex set.

Now suppose that the household preferences are given by some a utility function u : nR+

"

R. Then Corollary 5 implies that if u is strictly

quasi-concave, there is a unique bundle

x " B(p, I) that maximizes u : B(p, I) ! R.

The example is illustrated here.

2.2.6. Convex and Quasiconvex Functions

If we reverse the inequality sign in the definitions of concave and quasi-concave functions we obtain convex and quasi-convex functions. Convex Functions: f: D R! is a convex function iff for all x1 and x2 in D,

f(xt) ! tf(x1) + (1-t)f(x2), for all t! [0,1].

Convexity ! the region above the graph – set (x,y) or (D,R) is a convex set.

Strictly Convex Function: f: D R! is a strictly convex function iff for all x1 and x2 in D,

f(xt) < tf(x1) + (1- t) f(x2), for all t! (0,1).

Theorem 6: (a) f is a (strictly) convex function if and only if %f is a (strictly) concave fnctn.

(b) f is a (strictly) quasi-convex function if and only if %f is a (strictly) quasi-concave fnction.

+"

,"

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

25

!

Theorem 6 allows us to easily translate all of our propositions for concave and quasi-concave

functions to the analogues for convex and quasi-convex functions, which are provided here

for easy reference. !

• Suppose f1,...,fn are convex functions. Then for any *1,...,*n, for which each *i " 0, then

f-(*ifi is also a convex function. If, in addition, at least one fj is strictly convex and *j > 0,

then f is strictly convex.

• A linear function is both concave and convex.

• A (strictly) convex function is (strictly) quasi-convex.

• Suppose " : X ! R is quasi-convex and / : "(X) " R is nondecreasing.

Then (. 1 " ): X " R is quasi-convex. If " is strictlyquasi-convex and / is strictly increasing,

then (. 1 "!)is strictly quasi-convex.

• A function " : X " R is quasi-convex if and only if for each x - X, W(x) is convex.

• Suppose " : X " R attains a minimum on X. (a) I0 " is quasi-convex, then the set of

minimizers is convex. (b) If " is strictly quasi-convex, then theminimizer of " is unique.

Theorem. Points On & Above the Graph of a Convex Functn Always Form a Convex Set

Let D nR! be a convex set, R ! R. Let A* = {(x,y) | x ! D, f(x) ! y} be the set of points

“on and above” the graph of f: D!R.

f is a convex function ! A* is a convex set.

Proof: Sufficiency: We need to prove that f is a convex function ! A* is a convex set.

Consider the two points (x1,y1) and (x2,y2) in set A*.

Take the convex combination of the two points, we have: (xt,yt), in which:

xt = t x1 + (1-t) x2 and yt = t f(x1) + (1-t) f(x2)

We need to prove that (xt,yt) is also in set A* ! f(xt) ! yt.

According to the definition of a convex function: f(xt) ! tf(x1) + (1-t)f(x2) = yt

! f(xt) ! yt ! (xt,yt) is in set A* as well.

Necessity: We need to prove that A* is a convex set ! f is a convex function.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

26

Consider the two points on the graph of f (x): (x1,f(x1) and (x2,f(x2)).

A* is a convex set ! f(xt) ! tf(x1) + (1-t)f(x2) ! is a convex function.

Quasiconvex and Strictly Quasiconvex Functions

1. A function f: D!R is Quasiconvex iff for all x1 and x2 in D,

f(xt) ! max[f(x1), f(x2)] for all t ! [0,1].

2. A function f: D ! R is Strictly Quasiconvex iff for all x1 and x2 in D,

f(xt) < max [f(x1), f(x2)] for all t ! (0,1).

Theorem: Quasiconvexity and the Inferior Sets

f: D ! R is a quasiconvex function iff I(x) – Inferior set is a convex set for all x!D.

Proof: Sufficiency: We need to prove that f(x) is a quasiconvex ! I(x) Inferior set is a

convex set.

We need to prove that (xt,yt) is also in the I(x) set.

Consider the two points in I(x) set (x1,y1) and (x2,y2). Without loss of generality, we

assume that f(x1) ! f(x2) ! I(x1) ! I(x2) and f(x1)-f(x2) ! 0

f(x) is quasiconvex ! f(xt) ! max[f(x1),f(x2)]

Because f(x1) ! f(x2) ! f(x2) = max[f(x1), f(x2)]

! f(xt) ! f(x2) ! (xt,yt) ! I(x2) ! (xt,yt) ! A*.

Necessarity: We need to prove that I(x) is a convex set ! f(x) is a quasiconvex function.

We need to prove that f(xt) ! max[f(x1),f(x2)].

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Consider the two points in I(x): x1(x11, x1

2) and x2(x22,x2

2)

I(x) is a convex set ! the convex combination of the two points is also in I(x).

Without loss of generality, we assume that f(x1) ! f(x2) !max[f(x1), f(x2)] = f(x2).

f(xt) = t f(x1) + (1-t) f(x2) = f(x2) + t [f(x1)-f(x2)] ! f(x2) due to f(x1) ! f(x2).

! f(xt) ! max[f(x1), f(x2)] ! f(x) is a quasiconvex function.

Theorem: Concave/Convex and Quasiconcave/Quasiconvex Funtions

1. f(x) is (strictly) concave function iff –f(x) is (strictly) convex function.

2. f(x) is (strictly) quasiconcave function iff –f(x) is (strictly) quasiconvex function.

Summaries:

1. f concave ! convex sets beneath the graph.

2. f convex ! convex sets above the graph.

3. f quasiconcave ! Superior sets are convex sets.

4. f quansiconvex ! Inferior sets are convex sets.

5. f concave ! f quasiconcave.

6. f convex ! f quasiconvex.

7. f (strictly) concave ! -f (strictly) convex.

8. f (strictly) quasiconcave ! -f (strictly) quasiconvex.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

28

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3.1. Mathematical operation:

We are familiar with the operations of ordinary addition, multiplication of numbers. We have further studied the notions of union and intersection of sets. These are denoted by

a + b= c, a.b = d, A U B= C, A ! B= D.

In each of these situations an element (c, d, C or D) is assigned to the ordered pair of original elements as an outcome of operation considered. By an ordered pair we mean a pair of objects, say (a, b), which are considered in the order first a, then b.

Let us now consider a set S = {a, b, c, d, .. . . . } and we define an operation, given by the symbol *, (or o, or . , or any other symbol) which operating on an ordered pair of two elements of the set S gives the next element of the set as its outcome. For example, a * b= c,

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29

b * c= d, and so on. The rules or laws of such operation may be given in the form of postulates, theorems or any other definition. In general, an operation or composition is a set of rules given in the form of postulates, theorems or any definition which assigns to each ordered sub-set of a set some uniquely detennined element of that set.

Thus a mathematical system involves

(i) a non-empty set of elements ; (ii) one or more operations which associate with each ordered sub-set of the set, an

element also of the set; (iii) a set of operations or compositions, called postulates or axioms, which the

elements and the operations satisfy.

3.2. Binary composition. Operations are classified according to the number of elements of the set thatare involved in it, as unary, binary, ternary, ....... , n-ary, and so on.

Extraction of square root is a unary operation in arithmetic; for, only one element of the number system is operated upon.

An operation o (or *, or " or any other symbol) which when applied to two elements of a set S gives a unique element, also of the same set, is called a binary operation. Thus, if a!S and b!S and a unique element, c, also of the same set, exists such that c= a o b then the operation o is a binary operation in S. Given a set A, any mapping of AxA into A is called a binary composition on A.

Let o be a binary composition on A. By definition, mapping of A!A into A, i.e., (a, b) ! A!A, o(a, b)!A.

Addition is a binary operation on the set of even numbers, since the Sum of two even natural numbers is an natural number; but is not a binary operation on the set of odd natural numbers; since the sum of two odd natural numbers is an even natural number. Subtraction is a binary operation the set of all integers, positive and negative, but not, if be limited' to only positive integers.

Neither addition nor multiplicationis a binary operation on the set S= {0, 1, 2, 3, 4, 5}, since 3 + 4= 7 !S and 2 x3= 6 !S.

Let M2 be the set of all 2x2 matrices whose elements are rational numbers. For any two

matrices A, B! M2 , where A= 11 12

21 22

a aa a

! "# $% &

, B= 11 12

21 22

b bb b! "# $% &

, we have the usual sum A + B=

11 11 12 12

21 21 22 22

a b a ba b a b

+ +! "# $+ +% &

, which also belongs to M2. So the usual matrix addition is a binary

composition on M2.

If o be a binary operation defined over a set S, then two elemlents a and b of S, a 0 b ! S; this is said to be the closure property of the binary operation and the set S is said to be closed with

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30

respect to the operation o. A non-empty set together with one or more than one operations is called algebraic structure. For examples, (N, +), (Z, +, -), (R, +, '). are algebraic structures.

3.3. Laws of binary composition

Associative law: A binary operation o on the elements of the set S is said to be associative, if and only if, for any three elements a, b, c!S,

a o ( b o c ) = (a 0 b ) 0 c. For example, multiplication is an associative composition in the set of integers,

since (3 x 4) x 5= 3 x (4 x 5). Let the binary operation o be defined on R by a 0 b= a + 2b; a, b!R. Now ( a 0 b ) 0 c = (a + 2b) 0 c = a + 2b + 2c; a, b, c!R. But a o (boc)= a o (b +2c)= a + 2(b+ 2c) = a +2b+4c. Therefore the binary operation o as defined here is not associative.

Commutative law : A binary operation o on the elements of the set S is commutative, if and only if, for any two ,a b!S, a o b= b o a. For example, union and intersection of sets are commutative compositions in the sets of all sub-sets of a set, since A U B= B U A and A!B= B! A. Let o be the binary operation on R, that is, 0 : R R R! "

defined by o : ( , )x y x y! " ; x, y!R. Then o (6,2) = 6 -2 = 4 but o (2, 6)= 2! 6=! 4. Hence subtraction of real numbers is

not a commutative operation. .

Distributive law : Two binary operations o and *, say, on elements of the set S are said to be distributive, if and only if, for every element a, b and c !S,

a o (b * c)= (a o b)*(a o c) [left] (b * c) o a= (b o a)*(c o a) [right]

For example, multiplication is left as well as right distributive over addition in the set of real numbers. We say that the multiplication composition ( • ) distributes addition composition (+) for the set of real numbers. But the addition composition does not distribute the multiplication position in R, since a, b, c!R,

a • (b + c)= a • b + a • c but a + (b • c)! (a + b) • (a + c). Identity element. A set S is said to possess an identity element with respect the binary operation o, if there exists an element i! S with property that

I o x = x o I = x, for every x! S.

Here (S, o) is an algebraic structure with identity element. If i o x= x, then i is called the left identity element for the operation o and if x 0 i= x, then i is called the right identity element for o. If the set contains an element which is both a right and a left identity element for an operation, then we call it an identity element for that operation.

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31

Consider any element x of the set Q of rational numbers with respect to the binary operation addition. Obviously, zero is the identity element, since 0 + x= x + 0= x, for every x!Q.

1 is the identity element of Q for the binary operation mul-tiplication, since 1.x= x.1= x, for every x!Q.

It is easily seen that for the set N of natural numbers2 there is no identity clement for addition; but 1 is an identity element with respect to multiplication.

There is no need for an identity element to exist as we see that there is no such element for the ordinary operation of sub-traction in R, because this would have to be a number i such that

i x x i x! = ! = for all x, in the real number set R, which is impossible.

3.4. GROUP THEORY:

Theory of groups is one of the most important fundamental concepts of Modem Algebra. Group is, an algebraic structure with only binary opetation.

3.4.1. Groupoid, semi-group and monoid:

Consider the algebraic structure (S, 0) in which S is a non-empty set on which the binary composition 0 is defined. Thus S is closed with the operation o. Thus, if a, b ! S, then a 0 b ! S. Such a system consisting of a non-empty set S and a binary composition defined in S is called a groupoid.

If N be the set of natural numbers, then (N, +) is a groupoid because the set N is closed under addition. But the set of integers is not a groupoid under addition composition. The set S= { -2, -1, 0, 1, 2} is not a groupoid with respect to addition, since 2 + 2 = 4 !S, that is, the set S is not closed with addition composition.

If the binary composition 0 be commutative in S, then groupoid is said to be a commutative groupoid.

If a 0 x= x for all x!S, then a!S is called the left identily element of the groupoid. Similarly a !S is a right identity element of the groupoid, if x 0 a= x for all x!S .

The groupoid (Z, -) has no left identity element but zero is a right identity element of it. If a groupoid possesses a right as well as a left identity, then they are equal.

A system consisting of a non-empty set S and an associative binary composition in S is called a semi-group.

Consider, for example, a, b, c !Z, the set of all integers. Then we have (a + b) + c = a + (b + c) and (a . b). c= a. (b. c) . Thus Z being closed with the binary operations of addition and multiplication the systems (Z, +), and (Z, .) are semi-groups as those two compositions are associative in Z. But the system (Z, -) is not a semi-group, since subtraction does not satisfy the associative law.

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32

A system consisting of a non-empty set S and an associative binary composition with identity is called a monoid. Thus monoid is an associative groupoid with an identity element. If Z be the set of all integers, then (Z, +) is a monoid with identity element 0 and (Z, .) is a monoid with 1 as the identity element.

3.4.2. Group. A non-empty set S of elements a, b, c, ....... forms a group with respect to the binary operation *, if the following properties (axioms) hold:

(i). For every pair a and b!S, a * b is in S (closure law).

(ii). For any three elements a,b, c!S, a* (b * c)= (a * b) * c holds (associative law).

(iii) ! in S an element i, called a left identity, so that i * a= a, for every a!S.

iv) For each a in S, the equation x * a = i has a solution x !S.

This solution x is called a left inverse of a.

If, in addition to these postulates for a group; a * b= b * a (commutative law)

for all a and b in the group, the group is called a commutative or abelian group; otherwise it is called a non-abelian group.

For example, consider the set of integers (positive, negative and Zero) Z= {. ........ , -2, -1, 0, 1,2, ...... } on which the binary operation addition is applied.

For any a,b,c !Z, we have

i. a + b! Z (closure) ii. (a + b) + c= a + (b + c) (associativity), iii. 0 + a = a (0 is the left identity element) iv. (-a )+a = 0 (left inverse of a is a Z! " )

Hence the set of integers forms a group with respect to addition, since it satisfies all the group axioms. Moreover, a+ b = b +a, which shows that it is an abelian group. Thus the set Z of all integers (positive, negative zero) with additive binary operation is an abelian group.

Consider, again, the set Q+ of non-zero rational numbers on which the binary operation multiplication is applied. For any a,b,c!Q+, we have

i. a ! b!Q+ (closure), ii. (a x b) x c = a x (b x c) (assoclativity), iii. 1 x a = a (1 is the left identity element), iv. 1 1aa! = (1 a is the left inverse of a).

Moreover, a x b= b x a.

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33

Hence the set Q+ of non-zero rational numbers is an abelian group with respect to multiplication. If Q be the set of rational numbers, then (Q, +) is an abelian group but (Q,.) is not, since 0 has no left inverse in Q.

A group with addition binary operation is known as additive group and that with multiplication binary operation is known as multiplicative group. A group (or the binary operation * is often written as (G, *) or < G, * >, where G refers to the set forming the group. If the underlying set in it group consists of a finite number of elements, then it is called a finite group; the number of elements in the underlying set determines its order. An infinite group consists o( infinite number of elements in'its underlying set. It is said to be of order zero or of infinite order.

Generally the order o( the group G is denoted by a(G). Sometimes only the set symbol G is used to denote a group in context of a given operation.

3.4.3. Rings: A set of elements a, b, c, ...... forms a ring R with respect to the binary compositions --addition and multiplication, defined on R, if

i. the set forms a commutative group with respect to addition, ii. the set is closed with respect to multiplication, iii. the associative law a.(b.c) = (a.b).c holds for multiplication, and iv. the distributive laws (right as well as left)

a.(b + c) = ab + a.c and (b + c).a ... b.a + c.a hold.

This algebraic structure is often written as (R, + , • ) or < R, + , • > where R is the non-empty set of the ring. When the operations + and 2 are defined, the set R defines a ring.

A ring R in which multiplication is commutative is commutative ring. For a commutative ring R, a .b= b . a , for all a, b !R. The set of all integers with two binary operations, and multiplication forms a ring.

Corrolary. Since the ring is a group with respect to addition, all the group properties hold for addition.

The unique identity of the additive group, denoted by ordinary symbol for the number zero, is called the zero element of the ring and is denoted by O. Thus a + 0 = 0 + a = a, for every a in the ring.

The unique additive inverse of an element a is denoted by (- a), such that a + (- a)= 0= (- a) + a.

For the additive operation, left as we.ll as right cancel" laws hold good in a ring; that is, for an a, b, e !R,

a + b = a + c implies b = c and b + a = c + a implies b = c.

There will be a unique solution x= b + (- a), .written as (b - a), of the equation a + x = b. 3.4.4. Fields.

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34

A ring containing at least two elements is called an integral domain, if it be commutative, has a unity element and be without zero divisor. The ring of integers (Z, +, .) is an integral domain. The ring of even integers does not contain the unity element and hence it is not an integral domain, although it is without zero divisor. The set of natural numbers N does not form an integtal domain. Note. Some authors do not demand a unity element for an integral domain. Any ring containing at least two elements is called a field,if it be commutative, has a unity element and be such that all non-zero elements ,have multiplicative inverse. Thus an integral domain is a field, if every element a (! 0) has a multiplicative inverse a -1, such that a -1. a= unity element. The ring of rational numbers is a field, since it is a commutative ring with unity and each non-zero element has a multiplicative invese. The ring of all integers is not a field, as all non-zero elements in the ring of integers do not have multiplicative inverses. Thus a set F having at least two elements which algebraic structure (F, + , .) with the binary operations of addition and multiplication is called a field if the following be satisfied :

i. a + b! F, whenever a, b!F. ii. a + (b + c)= (a + b)+ c ; a, b, c !F. iii. There exists an element 0 !F such that a + 0= 0 + a = a, for all a!F. iv. a + b = b + a. v. a + x = b is solvable, for all a, b!F. vi. a . b!F, whenever a, b!F. vii. a. (b. c)= (a. b) . c, for all a, b, c !F. viii. There exists an element 1 !F such that a . 1= 1 . a= a, for all a !F. ix. a . b = b. a, for all a, b !F. x. a . x = b is solvable, for all a, b! F, where a! 0. xi. a. (b + c) = a. b + a. c . for all a, b, c !F. xii. (b + c) . a= b . a + c . a It follows from (x) that, for every a!F, a! 0, there exists an element a-I !F such that a-I . a= a . a-I = 1.

4.1. Vector Spaces: Let S be any non-empty set. If a * b !S for all a, b!S and a * b is unique, then * is

said to be an internal composition [e.g. addition of vectors] in the set S. This is a map

S S S! " .

Let vector V and a field F be two non-empty sets. If a o * !V for all a!F and *!V

and a o * is unique, the o is said to be an external composition [e.g. multiplication of vectors

by scalars] in V over F. This is a map F V V! " .

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35

Definition: Let (F, +, . ) be a field. Then a non-empty set V is called a vector space over

the field F, if in V there be defined an internal composition * and an external composition o

over F such that for all a, b!F for all ,, - !V,

(i) (V, *) is an abelian group.

(ii) a o [* o +] =[ a o * ] * [a o + ],

(iii) [a + b] o * =[a o * ] * [b o * ],

(iv) [a . b] o * = a o [b o * ]

(v) 1 o * = * , the unity scalar 1!F.

1 is the multiplicative identity of the field F.

The vector space V over the field F is denoted by V(F) or simply V.

4.2. Linear independence and dependence of vectors: Let V be a vector space over a field F.

Any vector , is said to be a linear combination of the vectors *1, *2, ....., *n !V if

* = c1. *1 + c2. *2 + ...... + cn. *n

where the scalars c1 , c2 , ......, cn !F.

As V is a vector space, so by vector addition and scalar multiplication in V, we have ,!V.

For a fixed set of vectors *1, *2, ....., *n different linear combinations are obtained by choosing

different sets of scalars.

Thus (2, 3, 4) = 3(1, 1, 1) + (- 1, 0, 1)

and (2, 3, 4) = (-2, 1, 2) + 12

(2, 0, 3) +3(1, 2 1,3 6

)

Let V be a vector space over a field F. A finite sub-set { *1, *2, ....., *n } of vectors of V is said

to be linearly dependent, if there exist scalars c1 , c2 , ......, cn !F, not all of them zero, such

that c1. *1 + c2. *2 + ...... + cn. *n = 0 .

Let V be a vector space over a field F. A finite sub-set { *1, *2, ....., *n } of vectors of V is said

to be linearly independent, if every relation of the form

c1. *1 + c2. *2 + ...... + cn. *n = 0 , ci !F, i = 1 (1) n

implies that ci=0, for each i = 1 (1) n

An infinite set of vectors of the vector space V is said to be linearly independent, if its every

finite sub-set be linearly independent, otherwise it is linearly dependent.

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36

A null set is assumed to be linearly independent.

The results below follow directly from the definitions.

(a) If two vectors be linearly dependent, then one of them is a scalar multiple of the other.

(b) If the vectors *1, *2, ....., *n be linearly dependent, then one of them can be expressed

as a linear combination of the others.

(c) A system consisting of a single non-zero vector is always linearly independent .

(d) Every super-set of a linearly dependent set of vectors is linearly dependent.

(e) Any sub-set of linearly independent set of vectors is linearly independent.

(f) A set of vectors which contains the zero vector is necessarily linearly dependent.

4.3. Linear span. Let V be a vector space over the field F and S be any non-empty sub-set

of V. Then the linear span of S is defined as the set of all linear combinations of finite sets of

elements of S. It is denoted by L(S).

Thus we have

L(S)={ c1. *1 + c2. *2 + ...... + cn. *n : *i !S; ci !F, i=1, 2, …, n}.

L(S) is said to be generated or spanned by the set S and S is said to be the set of generators of

L(S).For example, the set L(S)= 1

2

00!

!" #$ %& &' () *& &+ ,- .

of all 2!2 diagonal matrices is the linear span of

1 00 0! "# $% &

and 0 00 1! "# $% &

.

In other words, the sub-space spanned by a non-empty sub-set S of a vector space V is the set

of all linear combination of vectors in S. Clearly S!L(S).

The null set generates the set of zero vector alone, that is, { 0 }= L(2).

Note 1. A vector V! " is in the sub-space of V generated by S, if it can be expressed as a

linear combination of a finite number of vectors in S over F.

Note 2. The sub-set containing the single element (1, 0, 0) of the space V3 in F generates the

sub-space which is the totality of the elements of the form (a, 0, 0). The sub-set {(1, 0, 0), (0,

1, 0) } of V3 in F generates sub-space which is the totality of elements of the form (a, b, 0).

4.4. Basis and dimension of a vector space.

Let V be a vector space over the field F and S be a sub-set of V(F) such that

(i) S is a set of linearly independent vectors in and

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37

(ii) L(S) = ! "V, that is, each vector , in V is a linear combination of the finite

number of elements of S, (S generates V), then S called the basis set or simply

basis of V.

A zero vector cannot be an element of a basis set, such a set is linearly dependentK"

Consider the set B= {(1, 0, 0), (0, 1, 0), (0, 0, 1)} in V3 over the real numbers. This set is

linearly independent. Also B spans V3 because any vector (a1, a2, a3) of V3 can be written as a

linear combination of the vectors of B, viz. (a1, a2, a3) = a1(1, 0, 0)+ a2(0, 1, 0) + a3(0, 0, 1).

Hence B is a basis for V3 over the real numbers. It is called the standard basis of R3.

Consider, again, the set D = {1, 1, 0), (1, 0, 1), (0, 1, 1) } in.V3, over the real numbers. The

relation a(1, 1, 0) + b(l, 0, 1) + c(0, 1, 1) = 0 = (0, 0, 0) implies a = b = c = 0, that is, D is

linearly independent.

Furthermore any vector (a1, a2, a3) of V3 can be written as a linear combination of the vectors

of D as (a1, a2, a3) = 12

(a1 + a2 - a3) (1, 1, 0)

+ 12

(a1 + a3 – a2 ) (1, 0, 1)

+ 12

(a2 + a3 – a1) (0, 1, 1).

Thus D is also a basis of V3 over the real numbers. This shows that the basis for a vector space

need not be unique.

The vector space V is said to be finite dimensional or finitely generated, if there exists a finite

sub-set, S of V such that L(S)=V. Otherwise the vector space is infinite dimensional. The null

vector space which has no basis is of finite dimension which is zero.

The number of elements in any basis set of a finite dimensional vector space V(F) is called the

dimension of the vector space and is denoted by dimV. Vn(F) is n-dimensional, if its basis

contains n elements.

The vectot space R3 is of dimension 3 as B= {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is a basis of R3.

Similarly for R2 it is 2 and for R4· it is 4 and so on. The vector space spanned by the vectors

(1, 0, ..... , 0), (0, 1, 0, …. , 0) .... (0, 0, ......, 0, 1) is denoted by Vn(R) or simply by Vn.

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38

Example 1. Express (3, 4, 5) as a linear combination of , = (1, 2, 3) , - = (2, 3,4) and 2 =

(4,3,2) in the vector space V, of real numbers.

Let (3,4, 5) = a(1, 2, 3) + b(2, 3, 4) + c(4, 3, 2) for a. b, c !R

= (a + 2b + 4c ; 2a + 3b + 3c ; 3a+4b + 2c).

By vector treatment, we have

a + 2b +4c = 3,

2a +3b +3c = 4,

3a + 4b + 2c = 5.

Solving these equations by Cramer's rule, we have2 a=1/2, b=3/4, c=1/4

Thus (3, 4, 5)= 1/2 (1, 2, 3) + 3/4 (2, 3, 4) + 1/4 (4, 3, 2).

Example 2. Show that the vectors (1, 2, 1), (2,1, 0), (1,-1, 2) form a basis of the vector V3,

over the field of real numbers.

We know that if V(F) be a finite dimensional vector space of dimension n, then any set

of n linearly independent vectors in V forms a basis of V.

Now the set { (1, 0, 0), (0, 1, 0), (0, 0, 1) } forms a basis for V3 over real numbers.

Hence its dimension is 3. If we can show that the set

S = { (1, 2, 1), (2, 1, 0), (1, -1, 2) }

is linearly independent, then S will also form a basis of V3.

We have a1 (1, 2, 1) + a2 (4, 1, 0)+ a3 (1, -1, 2) = (0, 0, 0)

=> (a1 + 2a2 + a3 ; 2a1 + 2a2 – a3 ; a1 + 2a3 )= (0, 0, 0).

Therefore a1 + 2a2 + a3 = 0

2a1 + 2a2 – a3 = 0

a1 – a3 = 0

The coefficient matrix is 1 2 12 1 11 0 2

! "# $%# $# $& '

= A (say).

det A= - 9 . Thus the rank of A is 3. Hence solving these equations, we have the only solution

a1 = a2 = a3 = 0 . Therefore the set S is linearly independent.

Hence it forms a basis for V3 in the field of real numbers.

"

"