Discrete Mathematics

Relation

Cartesian Product If A1, A2, …, Am are nonempty sets, then

the Cartesian Product of these sets is the set of all ordered m-tuples (a1, a2, …, am), where ai Ai, i = 1, 2, … m.

Denoted A1 A2 … Am = {(a1, a2, …, am) | ai Ai, i = 1, 2, … m}

Cartesian Product Example If A = {1, 2, 3} and B = {a, b, c}, find A

B A B = {(1,a), (1,b), (1,c), (2,a), (2,b),

(2,c), (3,a), (3,b), (3,c)}

Subsets of the Cartesian Product Many of the results of operations on sets

produce subsets of the Cartesian Product set

Relational database Each column in a database table can be

considered a set Each row is an m-tuple of the elements from

each column or set No two rows should be alike

Using Matrices to Denote Cartesian Product For Cartesian Product of two sets, you can use a

matrix to find the sets. Example: Assume A = {1, 2, 3} and B = {a, b, c}.

The table below represents A × B.

a b c

1 (1, a) (1, b) (1, c)

2 (2, a) (2, b) (2, c)

3 (3, a) (3, b) (3, c)

Cardinality of Cartesian ProductThe cardinality of the Cartesian Product equals the product of the cardinality of all of the sets:

| A1 A2 … Am | = | A1 | | A2 | … | Am |

Subsets of the Cartesian Product Many of the results of operations on sets

produce subsets of the Cartesian Product set

Relational database Each column in a database table can be

considered a set Each row is an m-tuple of the elements from

each column or set No two rows should be alike

Introduction 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:

Example A is a set of students and B is a set of

courses A relation R may be defined as “register

the course”

Paul Giblock R CSCI 2710

Danny Camper R CSCI 2710

Relation on a Single Set Example A is the set of all courses A relation R may be defined as the course

is a prerequisite CSCI 2150 R CSCI 3400 R = {(CSCI 2150, CSCI 3400), (CSCI 1710,

CSCI 2910), (CSCI 2800, CSCI 2910), …}

Matrix of a Relation We can represent a relation between two

finite sets with a matrix MR = [mij], where

1 if (ai, bj) R0 if (ai, bj) R

mij =

Example Using the previous example where A = {1, 2, 3}

and B = {a, b, c}. The matrix below represents the relation R = {(1, a), (1, c), (2, c), (3, a), (3, b)}.

a b c

1 1 0 1

2 0 0 1

3 1 1 0

Digraph of a Relation Let R be a relation on A We can represent R using a diagram

Each element of A is a circle called a vertex If ai is related to aj, then draw an arrow from

the vertex ai to the vertex aj

In degree means number of arrows coming into a vertex

Out degree means number of arrows coming out of a vertex

Representing a RelationThe following three representations depict the same relation on A = {1, 2, 3}.

R = {(1, 1), (1, 3), (2, 3), (3, 2), (3, 3)}

1 0 1

0 0 1

0 1 1

1

2

3

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

Partial Order RelationsLet X be a set and R a relation on X

R is a partial order on X if R is reflexive, anti-symmetric and transitive.

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)}

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.

Equivalence classes

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

Define [a] ={ xX | xRa }

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.

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

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’

E.g. Dom (f) = X = {a, b, c, d}, Range (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 Range (f) Y.

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

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. Range (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 Range (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.

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.