math boot camp - class #1 - royal holloway, university of london · 2018-10-05 · alex vickery...

Math Boot Camp - Class #1

Alex Vickery

Royal Holloway - University of London

25th September, 2017

Alex Vickery (Royal Holloway) Math Boot Camp - Class #1 25th September, 2017 1 / 47

Outline:Today’s Class

1 The Real Number System:Overview

2 The Concept of Sets:Overview:Relationships between Sets:Operations on Sets:Laws of Set Operations


The Real Number System: Overview






The Real Number System:Integers:

Whole numbers such as:

1, 2, 3, 4, 5, ... ⇒ positive integers (N) (1)

Negative counterparts:

−1,−2,−3,−4,−5, ... ⇒ negative integers (2)

The anomaly:

0 ⇒ neither positive or negative (3)

Combine (1), (2) and (3) ⇒ set of all integers - Z



The Real Number System:Integers:

The Set of All Integers on a Line

−4 −3 −2 −1 0 1 2 3 4 (Z)



The Real Number System:Fractions:

Integers do not exhuast all the possible numbers, we have fractions e.g:

2

3,

5

4,

7

3, ... (4)

Also have negative fractions:

−2

3, −5

4, −7

3, ... (5)

If placed on a ruler they would fall between the integers.

Together; (4) and (5) make up the set of all fractions.



The Real Number System:Rational Numbers:

Any number that can be expressed as a ratio of two integers is called arational number - Q.The common property of fractional numbers is that they are expressible asa ratio of two integers:

a

bwhere a and b are integers and b 6= 0 (6)

Integers are also rational:

any integer, n, can be considered as the ratio:n

1(7)

The set of all integers and the set of all fractions together form theset of all rational numbers - Q.



The Real Number System:Fractions:

−4 −3 −2 −1 0 1 2 3 4 (Z)

The Integer Numbers:

−4 −3 −2 −1 0 1 2 3 4

−10/4 −1/2 7/10 11/3

(Q)

The Rational Numbers:



The Real Number System:Irational Numbers:

Irrational numbers cannot be expressed as the ratio of a pair of integers.Example(s):

√2 = 1.4142 ⇒ nonrepeating, nonterminating decimal. (8)

π = 3.1415 ⇒ nonrepeating, nonterminating decimal. (9)

Being a Nonrepeating, nonterminating decimal is a characteristic of allirrational numbers.



The Real Number System:Real Numbers:

If each irrational number were placed on a ruler they would fall betweentwo rational numbers.

Just as fractions fill in the gaps between the integers on a ruler, theirrational numbers fill in the gaps between the rational numbers.

The result of this filling in process is a continuum of numbers knownas the set of all real numbers - R.

When the set R is displayed on a straight line we often refer to this line asthe real line.



The Real Number System:Real Numbers:

−4 −3 −2 −1 0 1 2 3 4

−10/4 −1/2 7/10 11/3

(Q)

The Rational Numbers:

−4 −3 −2 −1 0 1 2 3 4

−10/4 −1/2 7/10 11/3

(R)

The Real Numbers:

−π π−√2

√2



The Real Number System:Figure 2.1

Integers - Z Fractions

Rational Numbers - Q Irrational Numbers

Real Numbers - R

(Chiang and Wainright page 8.)



The Real Number System:Overview

Number Systems

Symbol Name Example

N Natural 1, 2, 3, 4, 5, ...

Z Integer ..., -2, -1, 0, 1, 2, ...

Q Rational ab where a and b are integers and b 6= 0

R Real The limit of a convergent sequence of rational numbers

C Complex a + bi where a and b are real numbers and i is the square root of -1


The Concept of Sets: Overview:






The Concept of Sets:Overview

A set is simply a collection of distinct objects.

These objects can be a group of (distinct) numbers, persons, food items,etc ...

The objects in a set are called the elements of the set.

Example:

Three integers: 1, 3 and 4 form the set S .

S is the name of the set.

1, 3 and 4 are the elements of the set S .



The Concept of Sets:Overview:

There are two alternative ways of writing a set: by enumeration and bydescription.

Our set S can by written by enumeration of the elements:

S = {2, 3, 4}

Let I denote the set of all positive integers; enumeration becomes difficult,instead we can describe the elements and write:

I = {x | x a positive integer}




A set with a finite number of elements, e.g the set S , is called a finite set.

A set with an infinite number of elements, e.g the set I , is called a infiniteset.

Finite sets are always denumerable (or countable) i.e. their elements canbe counted in sequence.

Infinite sets may be either denumerable or nondenumerable .




Membership in a set is indicated by the symbol ∈,(read as “is an element of ”.)

Thus for the sets S and I :

2 ∈ S : 3 ∈ S : 8 ∈ I : 9 ∈ I : etc ...

Therefore 8 /∈ S means “ 8 is not an element of S ”.

Using earlier notation, the statement “ x is some real number ” can beexpressed by:

x ∈ R


The Concept of Sets: Relationships between Sets:






The Concept of Sets:Relationships between Sets:

When two sets are compared with each other, several possible relationshipsmay be observed.

If two sets S1 and S2 contain identical elements:

S1 = {2, 7, a, f } and S2 = {2, a, 7, f }

Then S1 and S2 are said to be equal (S1 = S2).

Note the the order of appearance of the elements in a set is immaterial.




Another relationship between sets is that one may be a subset of anotherset.

If we have two sets:

S = {1, 3, 5, 7, 9} and T = {3, 7}

then T is a subset of S , because every element of T is also anelement of S .

More formally:

T is a subset of S if and only if: x ∈ T ⇒ x ∈ S




The symbols that characterise the relationship are:

T ⊂ S , read as: “T is contained in S or is a subset of ”,S ⊃ T , read as: “S includes T or is a superset of ”.

It is possible that two given sets are subsets of each other. When thisoccurs we can be sure that the two sets are equal:

S1 ⊂ S2 and S2 ⊂ S1 ⇐⇒ S1 = S2

Note that, whereas the symbol ∈ relates an individual element to a set,the symbol ⊂ relates a subset to a set.




Consider the set: A = {1, 2, 3}:

How many subsets can be formed from the set A?

Answer:

{1}, {2}, {3}, each individual element can count as a distinct subset of A,

{1, 2}, {1, 3}, {2, 3}, so can any pair, etc ...

{1, 2, 3} and ∅ the set A also counts a subset of itself.

Any subset that does not contain all the elements of A is called aproper subset of A.




Consider the set: A = {1, 2, 3}:

The smallest possible subset of A is a set that contains no elements. Thisset is called the null set or empty set, denoted by the symbol: ∅

If the null set is not a subset of A (∅ 6⊂ A), then ∅ must contain at leastone element x such that x /∈ A. But by definition the null set has noelements so we cannot say: ∅ 6⊂ A; hence, the null set is a subset of A.




The third possible type of set relationship is when two sets have nocommon elements at all.

In that case, the two sets are said to be disjoint.

Example:The set of positive integers and the set of negative integers aremutually exclusive; thus they are disjoint sets.

The fourth type of relationship occurs when two sets have some elementsin common but some elements peculiar. In that event, the two sets areneither equal nor disjoint; also, neither is a subset of each other.


The Concept of Sets: Operations on Sets:






The Concept of Sets:Operations on Sets:

Three principle operations to consider here are the union, intersection andcompliment of sets.

To take the union of two sets A and B means to form a new setcontaining those elements (and only those elements) belonging to A, or toB, or to both A and B.

The union set is symbolized by A ∪ B (read:“ A union B ”).

If A = {3, 5, 7} and B = {2, 3, 4, 8} then:

A ∪ B = {2, 3, 4, 5, 7, 8}




Using what we learned earlier (and referring to fig 2.1) we see that theunion of the set of all integers (Z) and the set of all fractions is the set ofall rational numbers (Q).

Z ∪ (set of all fractions) = Q

Similarly, the union of the rational-number set (Q) and theirrational-number set yields the set of all real numbers (R).




The intersection of two sets A and B, on the other hand, is a new setwhich contains those elements (and only those elements) belonging toboth A and B.The intersection set is symbolized by: A ∩ B (read:“A intersection B ”).

If A = {3, 5, 7} and B = {2, 3, 4, 8} then:

A ∩ B = {3}

If A = {−3, 6, 10} and B = {9, 2, 7, 4} then:

A ∩ B = ∅

In the second case A and B are disjoint; the intersection is empty - noelement is common to A and B.




The operator symbols ∩ and ∪ have the connotations “and ” and “or ”respectively.

This is better appreciated when comparing the formal definitions below:

Intersection: A ∩ B = {x | x ∈ A and x ∈ B} (10)

Union: A ∪ B = {x | x ∈ A or x ∈ B} (11)




To explain the compliment of a set, lets first introduce the concept of theuniversal set.

Suppose that the universal set, U, is the set containing the first sevenpositive integers, U = {1, 2, 3, 4, 5, 6, 7}.

Given a set A = {3, 6, 7}, we can define another set A (read:“thecompliment of A ”) as the set that contains all the numbers in theuniversal set, U, that are not in the set A, that is:

A = {x | x ∈ U and x /∈ A} = {1, 2, 4, 5}




U

A

Set - A




U

A

A

Complement - AA

A




U

A B




U

A B

A ∩ B




U

A B

U

A B

A ∪ B


The Concept of Sets: Laws of Set Operations






The Concept of Sets:Laws of Set Operations:

From the previous figure it may be noted that the shaded area representsnot only A ∪ B but also B ∪ A.

Analogously, it was also clear the the black filled area representing A ∩ Balso represents B ∩ A.

When formalized, this result is known as the commutative law (of unionsand intersections):

A ∪ B = B ∪ A (12)

A ∩ B = B ∩ A (13)




To take the union of three sets A,B and C , we first take the union of anytwo sets and then “union” the resulting set with the third (the procedureis also applicable to the intersection operation ).

The results of such operations are presented on the following slides.

It is interesting that the order in which the sets are selected for operationis immaterial.




X

A C

B




X

A C

B

A ∪ B ∪ C




X

A C

B

A ∩ B ∩ C




This fact gives rise to the associative law (of unions and intersections):

A ∪ (B ∪ C ) = (A ∪ B) ∪ C (14)

A ∩ (B ∩ C ) = (A ∩ B) ∩ C (15)

These are strongly reminiscent of the algebraic laws:a + (b + c) = (a + b) + c and a× (b × c) = (a× b)× c .




There is also a law of operation that applies when unions and intersectionsare used in combination. This is the distributive law (of unions andintersections):

A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C ) (16)

A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C ) (17)

This resembles the algebraic law: a× (b + c) = (a× b) + (a× c).




Exercise:

Verify the distributive law given: A = {4, 5}, B = {3, 6, 7}, C = {2, 3}.

LHS: A ∪ (B ∩ C ) = {4, 5} ∪ {3} = {3, 4, 5}

RHS: (A ∪ B) ∩ (A ∪ C ) = {3, 4, 5, 6, 7} ∩ {2, 3, 4, 5} = {3, 4, 5}

Since both sides yield the same result, the first part is verified.




Verify the distributive law given: A = {4, 5}, B = {3, 6, 7}, C = {2, 3}.continued ...

LHS: A ∩ (B ∪ C ) = {4, 5} ∩ {2, 3, 6, 7} = ∅

RHS: (A ∩ B) ∪ (A ∩ C ) = ∅ ∪∅ = ∅

Thus, the second part is verified.


math boot camp - class #1 - royal holloway, university of london · 2018-10-05 · alex vickery...

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