Q.1 DETERMINE WHETHER THE GIVEN BINARY OPERATION IS COMMUTATIVE, ASSOCIATIVE . a) fOR x,y belogs to R. DEFINE x.y = x^2 +y. b) for m,n belongs to Z . define mn = mn - m - n Q.2 determine whether the each of the following is a binary operation . a) for x, y belongs to R , define (x - y)/x^2 + y b) for m, n belongs to Z, define (m+n)/2 Solution a) x.y = x^2+y y.x = y^2+x Thus x.y is not equal to y.x => not commutative For associativity (a.b).c = (a^2+b).c = (a^2+b)^2 +c a.(b.c) = a.(b^2+c) = a^2+b^2 +c Not associative b) m.n = mn-m-n n.m = nm-n-m => m.n =n.m => commutative For associtivity (a.b).c = (ab-a-b).c = (ab-a-b).c - (ab-a-b) -c = abc-ac-bc-ab-a-b-c a(b.c) = a(bc-b-c) = a(bc-b-c) - a- (bc-b-c) = abc-ab-ac-a-bc-b-c Associative Q2 a) x,y belongs to R (x-y)/ x^2+y belongs to R , hence its a binary operation b) m,n belongs to Z (m+n)/2 need not to be belongs to Z For ex. m=2,n=5 Here m,n belongs to Z, but m+n/2 = 7/2 doesn\'t belongs to Z Hence not a binary operation.

1. Q.1 DETERMINE WHETHER THE GIVEN BINARY OPERATION IS COMMUTATIVE,
ASSOCIATIVE .
a) fOR x,y belogs to R. DEFINE x.y = x^2 +y.
b) for m,n belongs to Z . define mn = mn - m - n
Q.2 determine whether the each of the following is a binary operation .
a) for x, y belongs to R , define (x - y)/x^2 + y
b) for m, n belongs to Z, define (m+n)/2
Solution
a) x.y = x^2+y
y.x = y^2+x
Thus x.y is not equal to y.x => not commutative
For associativity
(a.b).c = (a^2+b).c = (a^2+b)^2 +c
a.(b.c) = a.(b^2+c) = a^2+b^2 +c
Not associative
b) m.n = mn-m-n
n.m = nm-n-m
=> m.n =n.m => commutative
For associtivity
(a.b).c = (ab-a-b).c = (ab-a-b).c - (ab-a-b) -c
= abc-ac-bc-ab-a-b-c
a(b.c) = a(bc-b-c) = a(bc-b-c) - a- (bc-b-c) = abc-ab-ac-a-bc-b-c
Associative
Q2
a) x,y belongs to R
(x-y)/ x^2+y belongs to R , hence its a binary operation
b) m,n belongs to Z
(m+n)/2 need not to be belongs to Z
For ex. m=2,n=5
Here m,n belongs to Z, but m+n/2 = 7/2 doesn't belongs to Z
Hence not a binary operation