Binary Operations

Definition: A binary operation on a nonempty set A is a mapping defined on AA to A, denoted by f : AA  A. Binary Operation

Ex1. (a) Let “+” be the addition operation on Z. +:ZZ  Z defined by +(a, b) = a+b Let “” be the multiplication on R. : RR  R defined by (a, b) = ab Binary Operation

Ex1. (b) :ZZ  Z defined by (x, y) = x+y1 (1, 1) = (2, 3) = (1, 1) = (2, 3) = Then “” is a binary operation on Z. ∆:ZZ  Z defined by ∆(x, y) = 1+xy ∆(1, 1) = ∆(2, 3) = Then “∆” is a binary operation on Z. Binary Operation

Ex1. (c) Let “÷” be the division operation on Z. Then ÷(1, 2)=½. (1, 2)ZZ , but ½Z. Thus “÷” is not a binary operation. If we deal with “÷” on R , then “÷” is not a binary operation, either. Because ÷(a , 0) is undefined. But ÷ is a binary operation on R{0}. Binary Operation

Ex2. The intersection and union of two sets are both binary operations on the universal set . Binary Operation

Definitions: If “” is a binary operation on the nonempty set A, then we say “” is commutative if x  y = y  x, x, yA. If x  (y  z) = (x  y)  z,  x, y, z  A, then we say that the binary operation is associative. Binary Operation

Ex3.(a) The Operations “+” and “” on Z are both commutative and associative. Binary Operation

Ex3. (b) But operation –:ZZZ defined by –(a, b) = a – b is not commutative. Since The operation “–” is not associative, either. Because Binary Operation

Ex4. (a) Let “” be the operation defined as Ex1(b) on Z, x  y = x+y1. Then “” is both commutative and associative. Pf: Binary Operation

Ex4. (b) Let “∆” be the operation defined as Ex1(b) on Z, x∆y = 1+xy. Then “∆” is commutative but not associative. Pf: Binary Operation

Definition: Let : AA  A is a binary operation on a nonempty set A and let B  A. If xyB, x, y B, then we say B is closed with respect to “”. Binary Operation

Ex5. (a) The set S of all odd integers is closed with respect to multiplication. (b) Define :ZZ  Z by x  y =x+ y. Let B be the set of all negative integers. Then B is not closed with respect to “”, Binary Operation

Definition: Let A be a nonempty set and let : AA  A be a binary operation on A. An element e A is called an (two side) identity element with respect to “” if ex = x = xe, xA. Binary Operation

Ex6. (a) The integer 1 is an identity w. r. t. “”, but not w. r. t. “+”. The number 0 is an identity w. r. t. “+”. (b) Let “” be the operation defined as Ex1(b) on Z, x  y = x+y 1. Then Binary Operation

Ex6. (continuous) (c) Let “∆” be the operation defined as Ex1(b) on Z, x∆y = 1+xy. Then the operation has no identity element in Z. Pf: Binary Operation

Definition: Let e be the identity element for the binary operation “” on A and a A. If b A such that ab = e (or ba = e) then b is called a right inverse (or left inverse) of a w. r. t. . If both a b = e = b a, then b (denoted by a1) is called an (two-side) inverse of a; a1 is called an invertible element of a. Binary Operation

Note: The identity e and the two-side inverse of an element w. r. t. a binary operation  are unique. Pf: Binary Operation

Ex7. Let “” be the operation defined as Ex1(b) on Z, x  y = x+y 1. Then (2–x) is a two-side inverse of x w. r. t. “”, xZ. Pf: Binary Operation

Ex8. (a) Give a binary operation on Z as follow. (a) x  y = x

Ex8. (b) (b) x  y = x+2y. This operation is neither associative, nor commutative. Pf: Binary Operation

Ex8. (b) (continuous) (b) x  y = x + 2y.This operation has no identity, thus no inverse. Pf: Binary Operation

Ex8. (c) (c) x  y = x + xy +y. Binary Operation