In (a) - (c) determine whether the definition of * does give a binary operation
ID: 2962836 • Letter: I
Question
In (a) - (c) determine whether the definition of * does give a binary operation on the set. If it is a binary operation, determine whether it is commutative and associative. Explain in detail
a) On Z, define * by a*b = a+b+5
b) On R, define * by a*b = a-b
c) On Z, define * by a*b =ab +3
Compute the following table so as to define a commutative binary operation * on S= {a, b, c, d}
* a b c d
a b a c
b a d a
c c d d a
d b c
In (a) - (c), determine whether the given map is an isomorphism of the first binary structure with the second. Explain in detail
Explanation / Answer
(a)
binary operation, commutative, associative
a*b = a+b+5 = b*a => commutative
(a*b)*c =(a+b+5) *c = a+b+c+10 = a*(b+c+5) = a*(b*c) => associative
(b)
binary operation, not commutative, not associative
a*b = a-b
b*a = b-a => not commutative
(a*b)*c = (a-b)*c = a-b-c
a*(b*c) = a*(b-c) = a-(b-c) = a-b+c
=> not associative
(c)
binary operation
a*b = ab +3 = b8a => commutative
a*(b*c) = a*(bc+3) = a(bc+3) +3 = abc +3a+3
(a*b)*c = (ab+3)*c = (ab+3)c+3 = abc +3c+3
=> not associative
(a)
not isomorphishm since it is not onto(n = 1 doesnt have an preimage)