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LEARNING WAR [LeaRnine OBJECTIVES OBJECTIVES (Empty relation is the relation R in X given by R= CX*X. (@)_ Universal relation is the relation R in_X given by R= X * xX. (ii) Reflexive relation R in X is a relation with (a, aeRy aeX. (iv) Symmetric relation R in X isa relation satisfying (a, b) € R implies (b, a) € R. () Transitive relation R in X is a relation satisfying (a, b) ¢ R and (b, c) R implies that (@, cyeR. (vi) Equivalence relation R in X is a relation which is reflexive, symmetric and transitive. (vii) Equivalence class [a] containing a € X for an equivalence relation R in X is the subset of X containing all elements b related to a. (vill) A function f: X > Y is one-one (or injective) if f(x4)* F(&q) => Xj =X X1% EX. (ix) A function f: X -» Y is onto (or surjective) if given any y ¢ Y,3 x ¢ X such that f(x) = y. (x) A function f: X + Y is one-one and onto (or bijective), if fis both one-one and onto. (xi) The composition of functions f: A> B and g: B-> Cis the function gof: A— C given by gof(x)=g (f(x) y xe A. (xii) A function f: X—> Y is invertible if g: ¥> X such that gof = I, and fog =Iy- (xiii) A function f: X — Y is invertible if and only if f is one-one and onto. (xiv) Given a finite set X, a function £: X + X js one-one (respectively onto) if and only if fis onto (respectively one-one). This is the characteristic property of'a finite set. This is not true for infinite set. (avy A binary operation * ona setA isa function * from A x Ato A. (xvi) An element ¢ ¢ X is the identity element for binary operation *: X x XX, ifatesa=e*ayaeX. (xvii) Av element a € X is invertible for binary operation * : X x XX, ifthere exists b « X such that a * b =e = b * a where, eis the identity for the binary operation *, The element b is called inverse of a and is denoted by a}. (xviii) An operation * on X is commutative ifa*b=b*a y a,b in X. (xix) An operation * on X is associative if(a*b)*c=a*(b*c) y a. b,cinX. > INTRODUCTION : Relation is a word used incommon language. In our daily life we come across many relations. For example : (i) Delhi is the capital of India. (ii) 5 is a divisor of 1. (ii) Shyam is a son of Sohan (iv) Triangle ABC is congruent to triangle DEF (v) The set B is a subset of the set A. The above examples speak of some kind of relation between two places, persons, numbers or between two elements of a set. In this chapter, we will study different types of relations and functions, composition of functions, invertible functions and binary operations. TYPESOF RELATIONS Empty relation : A elation R in aset A is called empty relation, if'no element of Ais related to any element ofA, ie, R=OC AXA. Universal relation : A relation R in a set A is called universal relation, if each element of A is related to every element ofA, ie, R-AXA. : Both the empty relation and the universal relation are some times called trivial relations. Reflexive relation + Arelation R defined on the set A, is said to be reflexive if each element of A is related to itself. In other words, R will be reflexive if(a, a) c KR y a A. If there is any element in A which is not related to itself, the given relation will not be reflexive. For the relation R to be reflexive (a,a)e R y a < A. But that does not meant that a may nol be related to other element of A. Inother words, a is essentially related to itself but it may also be related to other elements of A, whereas in case of identify relation, ais related to a and only to a, Hence every identity relation is reflective but every reflexive relation is not an identity relation. 
