R. Johnsonbaugh,

Discrete Mathematics 5th edition, 2001

Chapter 2

The Language of Mathematics

2.1 Sets

Set = a collection of distinct unordered objects

Members of a set are called elements How to determine a set

Listing: Example: A = {1,3,5,7}

Description Example: B = {x | x = 2k + 1, 0 < k < 3}

Finite and infinite sets Finite sets

Examples: A = {1, 2, 3, 4} B = {x | x is an integer, 1 < x < 4}

Infinite sets Examples:

Z = {integers} = {…, -3, -2, -1, 0, 1, 2, 3,…} S={x| x is a real number and 1 < x < 4} = [0, 4]

Some important sets

The empty set has no elements.

Also called null set or void set. Universal set: the set of all elements about

which we make assertions. Examples:

U = {all natural numbers} U = {all real numbers} U = {x| x is a natural number and 1< x<10}

Cardinality Cardinality of a set A (in symbols |A|) is the

number of elements in A Examples:

If A = {1, 2, 3} then |A| = 3

If B = {x | x is a natural number and 1< x< 9}

then |B| = 9

Infinite cardinality Countable (e.g., natural numbers, integers) Uncountable (e.g., real numbers)

Subsets

X is a subset of Y if every element of X is also contained in Y

(in symbols X Y)

Equality: X = Y if X Y and Y X

X is a proper subset of Y if X Y but Y X Observation: is a subset of every set

Power set The power set of X is the set of all subsets of X,

in symbols P(X), i.e. P(X)= {A | A X} Example: if X = {1, 2, 3},

then P(X) = {, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}

Theorem 2.1.4: If |X| = n, then |P(X)| = 2n.

Set operations:Union and Intersection

Given two sets X and Y The union of X and Y is defined as the set

X Y = { x | x X or x Y}

The intersection of X and Y is defined as the set

X Y = { x | x X and x Y}

Two sets X and Y are disjoint if X Y =

Complement and Difference The difference of two sets

X – Y = { x | x X and x Y}

The difference is also called the relative complement of Y in X

Symmetric difference

X Δ Y = (X – Y) (Y – X) The complement of a set A contained in a

universal set U is the set Ac = U – A In symbols Ac = U - A

Venn diagrams

A Venn diagram provides a graphic view of sets

Set union, intersection, difference, symmetric difference and complements can be identified

Properties of set operations (1)

Theorem 2.1.10: Let U be a universal set, and A, B and C subsets of U. The following properties hold:

a) Associativity: (A B) C = A (B C)

(A B) C = A (B C)

b) Commutativity: A B = B A

A B = B A

Properties of set operations (2)

c) Distributive laws: A(BC) = (AB)(AC) A(BC) = (AB)(AC)d) Identity laws: AU=A A = Ae) Complement laws: AAc = U AAc =

Properties of set operations (3)

f) Idempotent laws:

AA = A AA = A

g) Bound laws:

AU = U A = h) Absorption laws:

A(AB) = A A(AB) = A

Properties of set operations (4)

i) Involution law: (Ac)c = A

j) 0/1 laws: c = U Uc = k) De Morgan’s laws for sets:

(AB)c = AcBc

(AB)c = AcBc

2.2 Sequences and strings A sequence is an ordered list of numbers,

usually defined according to a formula: sn = a function of n = 1, 2, 3,...

If s is a sequence {sn| n = 1, 2, 3,…}, s1 denotes the first element,

s2 the second element,…

sn the nth element…

{n} is called the indexing set of the sequence. Usually the indexing set is N (natural numbers) or an infinite subset of N.

Examples of sequences

Examples:

1. Let s = {sn} be the sequence defined by

sn = 1/n , for n = 1, 2, 3,… The first few elements of the sequence are: 1, ½, 1/3, ¼,

1/5,1/6,…

2. Let s = {sn} be the sequence defined by

sn = n2 + 1, for n = 1, 2, 3,… The first few elements of s are: 2, 5, 10, 17, 26, 37, 50,…

Increasing and decreasingA sequence s = {sn} is said to be

increasing if sn < sn+1 decreasing is sn > sn+1,

for every n = 1, 2, 3,…

Examples: Sn = 4 – 2n, n = 1, 2, 3,… is decreasing:

2, 0, -2, -4, -6,…

Sn = 2n -1, n = 1, 2, 3,… is increasing:

1, 3, 5, 7, 9, …

Subsequences

A subsequence of a sequence s = {sn} is a sequence t = {tn} that consists of certain elements of s retained in the original order they had in s Example: let s = {sn = n | n = 1, 2, 3,…}

1, 2, 3, 4, 5, 6, 7, 8,…

Let t = {tn = 2n | n = 1, 2, 3,…} 2, 4, 6, 8, 10, 12, 14, 16,… t is a subsequence of s

Sigma notation

If {an} is a sequence, then the sum

m

ak = a1 + a2 + … + am

k = 1

This is called the “sigma notation”, where the

Greek letter indicates a sum of terms from the sequence

Pi notation

If {an} is a sequence, then the product

m

ak = a1a2…am

k=1

This is called the “pi notation”, where the

Greek letter indicates a product of terms of the sequence

Strings

Let X be a nonempty set. A string over X is a finite sequence of elements from X. Example: if X = {a, b, c} Then = bbaccc is a string over X Notation: bbaccc = b2ac3

The length of a string is the number of elements of and is denoted by ||. If = b2ac3 then || = 6.

The null string is the string with no elements and is denoted by the Greek letter (lambda). It has length zero.

More on strings

Let X* = {all strings over X including } Let X+ = X* - {}, the set of all non-null strings Concatenation of two strings and is the

operation on strings consisting of writing followed by to produce a new string Example: = bbaccc and = caaba,

then = bbaccccaaba = b2ac4a2ba

Clearly, || = | | + ||

2.3 Number systems Binary digits: 0 and 1, called bits. In this section we study: binary, hexadecimal

and octal number systems.Review of decimal system: Example: 45,238 is equal to

8 ones 8 x 1 = 8

3 tens 3 x 10 = 30

2 hundreds 2 x 100 = 200

5 thousands 5 x 1000 = 5000

4 ten thousands 4 x 10000 = 40000

Binary number system From binary to decimal: The number 1101011 is equivalent to

1 one 1 x20 = 1 1 two 1x21 = 2 0 four 0x22 = 0 1 eight 1x23 = 8 0 sixteen 0x24 = 0 1 thirty-two 1x25 = 32 1 sixty-four 1x26 = 64 107 in decimal base

From decimal to binary The number 7310 is equivalent to

73 = 2 x 36 + remainder 1 36 = 2 x 18 + remainder 0 18 = 2 x 9 + remainder 0 9 = 2 x 4 + remainder 1 4 = 2 x 2 + remainder 0 2 = 2 x 1 + remainder 0

7310 = 10010012

(write the remainders in reverse order preceded by the quotient)

Binary addition table

0 1

0 0 11 1 10

Adding binary numbers

Example: add 1001012 + 1100112

1 1 1 carry ones

1001012

1100112

10110002

Hexadecimal number system

Decimal system0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 1 2 3 4 5 6 7 8 9 A B C D E F

Hexadecimal system

Hexadecimal to decimal

The hexadecimal number 3A0B16 is

11 x 160 = 11

0 x 161 = 0

10 x 162 = 2560

3 x 163 = 12288

1485910

Decimal to hexadecimal

Given the number 234510

2345 = 146x16 + remainder 9

146 = 9x16 + remainder 2

234510 is equivalent to the hexadecimal number 92916

Hexadecimal addition

Add 23A16 + 8F16

23A16

+ 8F16

2C916

2.4 Relations Given two sets X and B, its Cartesian product

XxY is the set of all ordered pairs (x,y) where xX and yY In symbols XxY = {(x, y) | xX and yY}

A binary relation R from a set X to a set Y is a subset of the Cartesian product XxY Example: X = {1, 2, 3} and Y = {a, b} R = {(1,a), (1,b), (2,b), (3,a)} is a relation between X

and Y

Domain and rangeGiven a relation R from X to Y, The domain of R is the set

Dom(R) = { xX | (x, y) R for some yY}

The range of R is the set Rng(R) = { yY | (x, y) R for some x X}

Example: if X = {1, 2, 3} and Y = {a, b} R = {(1,a), (1,b), (2,b)} Then: Dom(R)= {1, 2}, Rng(R) = (a, b}

Example of a relation Let X = {1, 2, 3} and Y = {a, b, c, d}. Define R = {(1,a), (1,d), (2,a), (2,b), (2,c)} The relation can be pictured by a graph:

Properties of relations

Let R be a relation on a set X

i.e. R is a subset of the Cartesian product XxX

R is reflexive if (x,x) R for every xX R is symmetric if for all x, y X such that (x,y)

R then (y,x) R R is transitive if (x,y) R and (y,z) R imply

(x,z) R R is antisymmetric if for all x,yX such that

xy, if (x,y) R then (y,x) R

Order relationsLet X be a set and R a relation on X

R is a partial order on X if R is reflexive, antisymmetric and transitive.

Let x,yX If (x,y) or (y,x) are in R, then x and y are

comparable If (x,y) R and (y,x) R then x and y are

incomparable

If every pair of elements in X are comparable, then R is a total order on X

Inverse of a relationGiven a relation R from X to Y, its inverse R-1

is the relation from Y to X defined by

R-1 = { (y,x) | (x,y) R } Example: if R = {(1,a), (1,d), (2,a), (2,b), (2,c)}

then R -1= {(a,1), (d,1), (a,2), (b,2), (c,2)}

2.5 Equivalence relations

Let X be a set and R a relation on X R is an equivalence relation on X R is

reflexive, symmetric and transitive. Example: Let X = {integers} and R be the

relation on X defined by: xRy x - y = 5. It is easy to show that R is an equivalence relation on the set of integers.

Partitions

A partition S on a set X is a family {A1, A2,…, An} of subsets of X, such that A1A2A3…An = X Aj Ak = for every j, k with j k,

1 < j, k < n. Example: if X = {integers}, E = {even

integers) and O = {odd integers}, then S = {E, O} is a partition of X.

Partitions and equivalence relations

Theorem 2.5.1: Let S be a partition on a set X.

Define a relation R on X by xRy if x, y are in the same set T for T S. Then R is an equivalence relation on X.

i.e. an equivalence relation on a set X corresponds to a partition of X and conversely.

Equivalence classes

Let X be a set and let R be an equivalence relation on X. Let a X.

Define [a] ={ xX | xRa } Let S = { [a] | a X }

Theorem 2.5.9: S is a partition on X. The sets [a] are called equivalence classes

of X induced by the relation R. Given a, b X, then [a] = [b] or [a][b] =

Set of equivalence classes

If R is an equivalence relation on a set X, define X/R = {[a] | a X }.

Theorem 2.5.16: If each equivalence class on a finite set X has k elements, then X/R has |X|/k elements, i.e. |X/R| = |X|/ k.

2.6 Matrices of relations Let X, Y be sets and R a relation from X to Y Write the matrix A = (aij) of the relation as

follows: Rows of A = elements of X Columns of A = elements of Y Element ai,j = 0 if the element of X in row i and

the element of Y in column j are not related Element ai,j = 1 if the element of X in row i and

the element of Y in column j are related

The matrix of a relation (1)Example:

Let X = {1, 2, 3}, Y = {a, b, c, d}

Let R = {(1,a), (1,d), (2,a), (2,b), (2,c)}

The matrix A of the relation R is

A =

a b c d

1 1 0 0 1

2 1 1 1 0

3 0 0 0 0

The matrix of a relation (2) If R is a relation from a set X to itself and A is the

matrix of R then A is a square matrix. Example: Let X = {a, b, c, d} and R = {(a,a),

(b,b), (c,c), (d,d), b,c), (c,b)}. Then

A =

a b c d

a 1 0 0 0

b 0 1 1 0

c 0 1 1 0

d 0 0 0 1

The matrix of a relation on a set XLet A be the square matrix of a relation R from

X to itself. Let A2 = the matrix product AA. R is reflexive All terms aii in the main

diagonal of A are 1. R is symmetric aij = aji for all i and j,

i.e. R is a symmetric relation on X if A is a symmetric matrix

R is transitive whenever cij in C = A2 is nonzero then entry aij in A is also nonzero.

2.7 Relational databases

A binary relation R is a relation among two sets X and Y, already defined as R X x Y.

An n-ary relation R is a relation among n sets X1, X2,…, Xn, i.e. a subset of the Cartesian product, R X1 x X2 x…x Xn. Thus, R is a set of n-tuples (x1, x2,…, xn) where

xk Xk, 1 < k < n.

Databases

A database is a collection of records that are manipulated by a computer. They can be considered as n sets X1 through Xn, each of which contains a list of items with information.

Database management systems are programs that help access and manipulate information stored in databases.

Relational database model

Columns of an n-ary relation are called attributes An attribute is a key if no two entries have the

same value e.g. social security number

A query is a request for information from the database

Operators

The selection operator chooses n-tuples from a relation by giving conditions on the attributes

The projection operator chooses two or more columns and eliminates duplicates

The join operator manipulates two relations

2.8 Functions

A function f from X to Y (in symbols f : X Y) is a relation from X to Y such that Dom(f) = X and if two pairs (x,y) and (x,y’) f, then y = y’

Example:

Dom(f) = X = {a, b, c, d},

Rng(f) = {1, 3, 5}

f(a) = f(b) = 3, f(c) = 5, f(d) = 1.

Domain and Range Domain of f = X Range of f =

{ y | y = f(x) for some x X} A function f : X Y assigns to

each x in Dom(f) = X a unique element y in Rng(f) Y.

Therefore, no two pairs in f have the same first coordinate.

Modulus operator Let x be a nonnegative integer and y a positive

integer r = x mod y is the remainder when x is divided

by y Examples:

1 = 13 mod 3

6 = 234 mod 19

4 = 2002 mod 111

mod is called the modulus operator

One-to-one functions A function f : X Y is one-to-one for each y Y there exists at most one x X

with f(x) = y. Alternative definition: f : X Y is one-to-one

for each pair of distinct elements x1, x2 X there exist two distinct elements y1, y2 Y such that f(x1) = y1 and f(x2) = y2.Examples: 1. The function f(x) = 2x from the set of real numbers to itself is

one-to-one 2. The function f : R R defined by f(x) = x2 is not one-to-one,

since for every real number x, f(x) = f(-x).

Onto functions

A function f : X Y is onto

for each y Y there exists at least one x X with f(x) = y, i.e. Rng(f) = Y. Example: The function f(x) = ex from the set of real

numbers to itself is not onto Y = the set of all real numbers. However, if Y is restricted to Rng(f) = R +, the set of positive real numbers, then f(x) is onto.

Bijective functions

A function f : X Y is bijective

f is one-to-one and onto Examples:

1. A linear function f(x) = ax + b is a bijective function from the set of real numbers to itself

2. The function f(x) = x3 is bijective from the set of real numbers to itself.

Inverse function Given a function y = f(x), the inverse f -1 is the

set {(y, x) | y = f(x)}. The inverse f -1 of f is not necessarily a

function. Example: if f(x) = x2, then f -1 (4) = 4 = ± 2, not a

unique value and therefore f is not a function.

However, if f is a bijective function, it can be shown that f -1 is a function.

Exponential and logarithmic functions

Let f(x) = 2x and g(x) = log 2 x = lg x

f ◦ g(x) = f(g(x)) = f(lg x) = 2 lg x = x g ◦ f(x) = g(f(x)) = g(2x) = lg 2x = x

Therefore, the exponential and logarithmic functions are inverses of each other.

Composition of functions Given two functions g : X Y and f : Y Z,

the composition f ◦ g is defined as follows:

f ◦ g (x) = f(g(x)) for every x X. Example: g(x) = x2 -1, f(x) = 3x + 5. Then

f ◦ g(x) = f(g(x)) = f(3x + 5) = (3x + 5)2 - 1

Composition of functions is associative:

f ◦ (g ◦h) = (f ◦ g) ◦ h, But, in general, it is not commutative:

f ◦ g g ◦ f.

Binary operators

A binary operator on a set X is a function f that associates a single element of X to every pair of elements in X, i.e. f : X x X X and f(x1, x2) X for every pair of elements x1, x2.

Examples of binary operators are addition, subtraction and multiplication of real numbers, taking unions or intersections of sets, concatenation of two strings over a set X, etc.

Unary operators A unary operator on a set X associates to

each single element of X one element of X. Examples:

1. Let X = U be a universal set and P(U) the power set of U. Define f : P(U) P(U) the function defined by f (A) = A', the set complement of A in U, for every A U. Then f defines a unary operator on P(U).

String inverseLet X be any set, X* the set of all strings over X.

If = x1x2…xn X*, let f() = -1 = xnxn-1…x2x1, the string written in reverse order.

Then f :X* X* is a function that defines a unary operator on X*.

Observe that -1 = -1 =