discrete mathematics relation. cartesian product if a 1, a 2, …, a m are nonempty sets, then the...

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.