1. Binary Operations

2. 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

3. 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

4. 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

5. 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

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

7. 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

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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

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

19. 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

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

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

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

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