binary operations. binary operation definition: a binary operation on a nonempty set a is a mapping...
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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. (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. (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: