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Relation
If A and B are two non-empty sets, then a relation R from A to B is a subset of A x B. If R โ A x B and (a, b) โ R, then we say that a is related to b by the relation R, written as aRb. Domain and Range of a Relation Let R be a relation from a set A to set B. Then, set of all first components or coordinates of the ordered pairs belonging to R is called : the domain of R, while the set of all second components or coordinates = of the ordered pairs belonging to R is called the range of R. Thus, domain of R = {a : (a , b) โ R} and range of R = {b : (a, b) โ R} Types of Relations (i) Void Relation As ฮฆ โ A x A, for any set A, so ฮฆ is a relation on A, called the empty or void relation. (ii) Universal Relation Since, A x A โ A x A, so A x A is a relation on A, called the universal relation. (iii) Identity Relation The relation IA = {(a, a) : a โ A} is called the identity relation on A. (iv) Reflexive Relation A relation R is said to be reflexive relation, if every element of A is related to itself. Thus, (a, a) โ R, โ a โ A = R is reflexive. (v) Symmetric Relation A relation R is said to be symmetric relation, iff (a, b) โ R (b, a) โ R,โ a, b โ A i.e., a R b โ b R a,โ a, b โ A โ R is symmetric. (vi) Anti-Symmetric Relation A relation R is said to be anti-symmetric relation, iff (a, b) โ R and (b, a) โ R โ a = b,โ a, b โ A
chapter 1. Relations and Functions
(vii) Transitive Relation A relation R is said to be transitive relation, iff (a, b) โ R and (b, c) โ R โ (a, c) โ R, โ a, b, c โ A (viii) Equivalence Relation A relation R is said to be an equivalence relation, if it is simultaneously reflexive, symmetric and transitive on A. (ix) Partial Order Relation A relation R is said to be a partial order relation, if it is simultaneously reflexive, symmetric and anti-symmetric on A. (x) Total Order Relation A relation R on a set A is said to be a total order relation on A, if R is a partial order relation on A. Inverse Relation If A and B are two non-empty sets and R be a relation from A to B, such that R = {(a, b) : a โ A, b โ B}, then the inverse of R, denoted by R-1^ , i a relation from B to A and is defined by R-1^ = {(b, a) : (a, b) โ R} Equivalence Classes of an Equivalence Relation Let R be equivalence relation in A (โ ฮฆ). Let a โ A. Then, the equivalence class of a denoted by [a] or {a} is defined as the set of all those points of A which are related to a under the relation R. Composition of Relation Let R and S be two relations from sets A to B and B to C respectively, then we can define relation SoR from A to C such that (a, c) โ So R โ โ b โ B such that (a, b) โ R and (b, c) โ S. This relation SoR is called the composition of R and S. (i) RoS โ SoR (ii) (SoR)-1^ = R-1oS- known as reversal rule. Congruence Modulo m Let m be an arbitrary but fixed integer. Two integers a and b are said to be congruence modulo m, if a โ b is divisible by m and we write a โก b (mod m). i.e., a โก b (mod m) โ a โ b is divisible by m.
๏ท Division is not a binary operation on any of the sets N, Z, Q, R and C. However, it is not a binary operation on the sets of all non-zero rational (real or complex) numbers. ๏ท Exponential operation (a, b) โ ab^ is a binary operation on set N of natural numbers while it is not a binary operation on set Z of integers. Types of Binary Operations (i) Associative Law A binary operation * on a non-empty set S is said to be associative, if (a * b) * c = a * (b * c), โ a, b, c โ S. Let R be the set of real numbers, then addition and multiplication on R satisfies the associative law. (ii) Commutative Law A binary operation * on a non-empty set S is said to be commutative, if a * b = b * a, โ a, b โ S. Addition and multiplication are commutative binary operations on Z but subtraction not a commutative binary operation, since 2 โ 3 โ 3 โ 2. Union and intersection are commutative binary operations on the power P(S) of all subsets of set S. But difference of sets is not a commutative binary operation on P(S). (iii) Distributive Law Let * and o be two binary operations on a non-empty sets. We say that * is distributed over o., if a * (b o c)= (a * b) o (a * c), โ a, b, c โ S also called (left distribution) and (b o c) * a = (b * a) o (c * a), โ a, b, c โ S also called (right distribution). Let R be the set of all real numbers, then multiplication distributes addition on R. Since, a.(b + c) = a.b + a.c,โ a, b, c โ R. (iv) Identity Element Let * be a binary operation on a non-empty set S. An element e a S, if it exist such that a * e = e * a = a, โ a โ S. is called an identity elements of S, with respect to *. For addition on R, zero is the identity elements in R. Since, a + 0 = 0 + a = a, โ a โ R
For multiplication on R, 1 is the identity element in R. Since, a x 1 =1 x a = a,โ a โ R Let P (S) be the power set of a non-empty set S. Then, ฮฆ is the identity element for union on P (S) as A โช ฮฆ =ฮฆ โช A = A, โ A โ P(S) Also, S is the identity element for intersection on P(S). Since, A โฉ S=A โฉ S=A, โ A โ P(S). For addition on N the identity element does not exist. But for multiplication on N the idenitity element is 1. (v) Inverse of an Element Let * be a binary operation on a non-empty set โSโ and let โeโ be the identity element. Let a โ S. we say that a-1^ is invertible, if there exists an element b โ S such that a * b = b * a = e Also, in this case, b is called the inverse of a and we write, a-1^ = b Addition on N has no identity element and accordingly N has no invertible element. Multiplication on N has 1 as the identity element and no element other than 1 is invertible. Let S be a finite set containing n elements. Then, the total number of binary operations on S in nn Let S be a finite set containing n elements. Then, the total number of commutative binary operation on S is n [n(n+1)/2].
