Binary Operations

Binary Operation

Definition:

A binary operation on a nonempty set A is a mapping defined on AA to A, denoted by f : AA A.

Binary Operation

Ex1. (a) Let “+” be the addition operation on Z.

+:ZZ Z defined by +(a, b) = a+b

Let “” be the multiplication on R.

: RR R defined by (a, b) = ab

Binary Operation

Ex1. (b)

:ZZ Z defined by (x, y) = x+y1(1, 1) = (2, 3) =

Then “” is a binary operation on Z. ∆:ZZ 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)ZZ , but ½Z.

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, yA. 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 –:ZZZ 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+y1. 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 : AA A is a binary operation on a nonempty set A and let B A.

If xyB, 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 :ZZ 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 : AA A be a binary operation on A. An element e A is called an (two side) identity element with respect to “”

if ex = x = xe, xA.

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 ab = e (or ba = 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 a1) is called an (two-side) inverse of a;a1 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. “”, xZ.

Pf:

Binary Operation

Ex8. (a)

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

Binary Operation

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.