Unit II Discrete Structures

Relations and Functions

SE (Comp.Engg.)

Cartesian productsThe Cartesian product of set A and set B is denoted by AB and equals {(a, b)aA and bB}. The elements of AB are ordered pairs. The elements of A1A2…An are ordered n-tuples.AB=AB

• Ex . A={2, 3, 4}, B={4, 5} , C={x,y}• A B ={<2,4>,<2,5>,<3,4>,<3,5>,<4,4>,<4,5>}

Tree diagrams for the Cartesian product

Relations

• Any subsets of AB is called a binary relation from A to B. Any subset of AA is called a binary relation on A.

For finite sets A and B with A=m and B=n, there are 2mn relations from A to B.

• Example: Let A = {1, 2, 3, 4}. Which ordered pairs are in the relation R = {(a, b) | a < b} ?

• Solution: R = {

• Domain= set of first elements in the caresian product .• Range= set of second elements in the caresian product .

Domain={1,2,3}Range={2,3,4}

(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}

• Converse of a Relation A is given by the relation à such that the elements in the ordered pairs in A are interchanged.

• i.e if xAy then y à x.

Matrix Representation of a Relation

• MR = [mij] (where i=row, j=col) mij={1 iff (i,j) R and 0 iff (i,j) R}

• Ex: R : {1,2,3}{1,2} where x > y – R = {(2,1),(3,1),(3,2)}

100

110

321

21

RM

Graph Representation of a Relation

Properties of Relations

• A relation R on a set A is called reflexive if (a, a)R for every element aA.

• A relation on a set A is called irreflexive if (a, a)R for every element aA.

•A relation R on a set A is called symmetric if (b, a)R whenever (a, b)R for all a, bA.

•A relation R on a set A is called antisymmetric if

a = b whenever (a, b)R and (b, a)R.

•A relation R on a set A is called asymmetric if (a, b)R implies that (b, a)R for all a,bA.

• A relation R on a set A is called transitive if whenever (a, b)R and (b, c)R, then (a, c)R for a, b, cA.

• A={1,2,3,4,5,6,7,8,9}• Give examples of relation R, such that,I. R is not reflexive and not irreflexive.II. R is symmetric as well as antisymmetric.III. R is transitive but not symmetric and not

reflexive.IV. R is transitive, reflexive but not symmetric.

Equivalence Relations

• Any binary relation that is:ReflexiveSymmetricTransitive

Equivalence Classes

•Let R be an equivalence relation on a set A. The set of all elements that are related to an element a of A is called the equivalence class of a. •The equivalence class of a with respect to R is denoted by [a]R.

•If b[a]R, b is called a representative of this equivalence class.

Partition

•A partition of a set S is a collection of disjoint nonempty subsets of S that have S as their union. In other words, the collection of subsets Ai, iI, forms a partition of S if and only if (i) Ai for iI

• Ai Aj = , if i j

• iI Ai = S

• Let R be an equivalence relation on a set S. Then the equivalence classes of R form a partition of S. Conversely, given a partition {Ai | iI} of the set S, there is an equivalence relation R that has the sets Ai, iI, as its equivalence classes.

Ex: if R={<1,2>,<1,1>,<2,3>,<3,2><3,4>}then

S={<1,2>,<1,1>,<2,3>,<3,2><3,4>,<2,1>,<4,3>}

3

42

1

3

42

1

Warshall’s algorithm

Let R is a relation in a set with n elements represented by matrix MR.

Calculate the matrices W0, W1, . . . , Wn where MR = W0

Wk is given by [wij(k)] where

[wij(k)] = 1, if there exists a path from vertex i to j

in the corresponding digraph, such that all the intermediate vertices of this path are in the set {1,2,…,k} 0, otherwise.

Partial order Relation

• A relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive.

• A set S together with a partial ordering R is called a partially ordered set, or POSET and denoted by (S,R). A partial order R is also denoted as .

• The elements a and b of a poset (S, ) are called comparable if either a b or b a. Otherwise a and b are called incomparable.

• If (S, ) is a partial ordering set and every two elements of S are comparable, S is called a totally ordered or linearly ordered set.

• A totally ordered set is called a Chain.

Hasse Diagrams

• Given any partial order relation defined on a finite set, it is possible to draw the directed graph so that all of these properties are satisfied.

• This makes it possible to associate a somewhat simpler graph, called a Hasse diagram, with a partial order relation defined on a finite set.

• Start with a directed graph of the relation in which all arrows point upward. Then eliminate:

1. the loops at all the vertices,

2. all arrows whose existence is implied by the transitive property,

3. the direction indicators on the arrows.

• Let A = {1, 2, 3, 9, 19} and consider the “divides” relation on A:

• For all , , | for some integer .a b A a b b ka k

• For the poset ({1,2,3,4,6,8,12}, |)

S={2,3,5}, Hasse diagrams of (P(S), ) ⊆and

D30: Dn indicates the poset with set of all intergers that divide n and the relation divides.

Extremal Elements: Maximal

• An element a in a poset (S, ≤) is called maximal if no element b in S exists such that,

a ≤ b• If there is one unique maximal element a, it is

called the maximum element (or the greatest element)

Extremal Elements: Minimal

• An element a in a poset (S, ≤) is called minimal if no element b in S exists such that,

b≤ a• If there is one unique minimal element a, it is

called the minimum element (or the least element)

• Let (S, ≤) be a poset and let AS. If u is an element of S such that a ≤ u for all aA then u is an upper bound of A

• An element x that is an upper bound on a subset A and is less than all other upper bounds on A is called the least upper bound on A. We abbreviate it as lub.

• Definition: Let (S, ≤) be a poset and let AS. If l is an element of S such that l ≤ a for all aA then l is an lower bound of A

• An element x that is a lower bound on a subset A and is greater than all other lower bounds on A is called the greatest lower bound on A. We abbreviate it glb.

Give lower/upper bounds & glb/lub of the sets:

{d,e,f}, {a,c} and {b,d}

{d,e,f}

• Lower bounds: , thus no glb• Upper bounds: , thus no lub

{a,c}• Lower bounds: , thus no glb• Upper bounds: {h}, lub: h

{b,d}• Lower bounds: {b}, glb: b• Upper bounds: {d,g}, lub: d because d ≤ g

• Find all upper and lower bounds of the following subset of A: B1={a, b}; B2={c, d, e};

Find the LUB and GLB of B={6,7,10} for the following Hasse diagram.

• 1

• 2• 3 • 4

• 5 • 6 • 7 • 8

• 10• 9

• 11

Lattices

• A lattice is a partially ordered set in which every pair of elements has both – a least upper bound and– a greatest lower bound

Union, Intersection, Difference and Composition of Relations

• R: AB and S: AB

• R: AB and S: BC

}),(),(|),{( SyxRyxBAyxSR

}),(),(|),{( SyxRyxBAyxSR

}),(),(|),{( SyxRyxBAyxSR

}),(),(|),{( RyxSyxBAyxRS

}),(),(,|),{( ScbRbaBbCAcaSR

Example: Let D and S be relations on A = {1, 2, 3, 4}.D = {(a, b) | b = 5 – a}S = {(a, b) | a < b}

D = {(1, 4), (2, 3), (3, 2), (4, 1)}S = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}DS = { (2, 4), (3, 3), (3, 4), (4, 2), (4, 3), (4, 4)}

• Example: Let the relations R and S be represented by the matrices

011

111

101

SRSR MMM

001

110

101

SM

000

000

101

SRSR MMM

010

001

101

RM

• Let A = [aij] be an mk zero-one matrix and B = [bij] be a kn zero-one matrix.

• Then the Boolean product of A and B, denoted by AB, is the mn matrix with (i, j)th entry [cij], where

• cij = (ai1 b1j) (ai2 b2i) … (aik bkj).

• cij = 1 if and only if at least one of the terms(ain bnj) = 1 for some n; otherwise cij = 0.

• Let MA = [aij], MB = [bij] and MC = [cij] represent relations A, B, and C, respectively, and C = AB

• Then MC = MAMB

• MAB = MAMB

Functions

• For nonempty sets, A,B, a function, or mapping, f from A to B, denoted f:A→B, is a relation from A to B in which every element of A appears exactly once as the first component of an ordered pair in the relation.

• i.e domain(f)=A|f|=|A|Codomain=B

Properties of functions

• f: AB, is one-to-one or injective, if each element of B appears at most once as the image of an element of A.

Therefore AB.

f: AB, is one-to-one if and only if for all a1, a2A, f(a1)=f(a2) a1=a2.

• f: AB, is onto, or surjective, if range of f=Bi.e. , for all bB there is at least one aA with f(a)=B.

Therefore AB.

• f: AB, is one-to-one onto, or bijective, if f is both one-to-one and onto.

Therefore A=B.

.each for

)),((=))((

bygiven is ,:

by denoted function, composite

then the,: and : If

Aa

afgafg

CAfg

CBgBAf

( )( ) ( ( )) ( ) , ( ) , ( ) ,g f g f g a x gf x gf y 1 1 2 3 gf(4) = z

• A function f is called invertible if the converse of f is also a function. The converse is called inverse function represented by f- 1.

• The function is invertible if and only if it is one-to-one and onto, or bijective.

• Discrete numeric functions (numeric functions)

• The class of functions • whose domain is the set of natural numbers • whose range is the set of real numbers

Operations of numeric functions

Shifting

• let a be a numeric function and i a positive integer

function a is shifted i positions to the right

Find a5

Find a-7

Recurrence Relation

A recurrence relation is an infinite sequence a1, a2, a3,…, an,…

in which the formula for the nth term an depends on one or more preceding terms,

with a finite set of start-up values or initial conditions

Examples

• Fibonacci sequence

Initial conditions:

f1 = 1, f2 = 1

Recursive formula:

f n+1 = f n-1 + f n for n > 3

First few terms:

Compound interest

• Given

– P = initial amount (principal)– n = number of years– r = annual interest rate– A = amount of money at the end of n years

At the end of: 1 year: A = P + rP = P(1+r) 2 years: A = P + rP(1+r) = P(1+r)2

3 years: A = P + rP(1+r)2 = P(1+r)3

…

• Obtain the formula A = P (1 + r) n

ExamplesLinear homogeneous• an = 1.2 an-1 : degree 1

• fn = fn-1 + fn-2 : degree 2

• an = 3an-3 : degree 3

Non-linear homogeneous• an = a2

n-1 + an-2

• an = nan-1 - 2an-2

Linear non-homogeneous• an = an-1 + 2n

• hn = 2hn-1 + 1

• an = 3an-1 + n

Non-linear non-homogeneous• an = a2

n-1 + 2n

• an = n2 an-1 + n

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The pigeonhole principle

• Suppose a flock of pigeons fly into a set of pigeonholes to roost

• If there are more pigeons than pigeonholes, then there must be at least 1 pigeonhole that has more than one pigeon in it

• If k+1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects

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Pigeonhole principle examples

• In a group of 367 people, there must be two people with the same birthday– As there are 366 possible birthdays

• In a group of 27 English words, at least two words must start with the same letter– As there are only 26 letters

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Generalized pigeonhole principle

• If N objects are placed into k boxes, then there is at least one box containing N/k objects

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Generalized pigeonhole principle examples

• Among 100 people, there are at least 100/12 = 9 born on the same month

• How many students in a class must there be to ensure that 6 students get the same grade (one of A, B, C, D, or F)?

– The “boxes” are the grades. Thus, k = 5– Thus, we set N/5 = 6– Lowest possible value for N is 26