Question

Let Q be the set of rational numbers and * be the binary operation on Q defined by a+b=ab/4 for all a, b in Q.
a.) what is the identify element of * on Q?
b.) Find the inverse of element of 'a' under * on Q?
c.) show that (a*b)*c=a*(b*c) for all a, b, c in Q.​

Answer 1

Solution :

Given binary operation :

* : Q → Q , a*b = ab/4

a.) Identify element :-

Let e be the identity element for *

Thus ,

=> a*e = a

=> ae/4 = a

=> e/4 = 1

=> e = 4 (unique)

Hence ,

Identify element is 4 .

b.) Inverse element :-

Let a' be the inverse element for *

Thus ,

=> a*a' = e

=> aa'/4 = 4

=> a' = 4×4/a

=> a' = 16/a

Hence ,

Inverse element is 16/a , a € Q .

c.) (a*b)*c = a*(b*c)

=> LHS = (a*b)*c

=> LHS = (ab/4)*c

=> LHS = [ (ab/4)c ] / 4

=> LHS = abc / 4×4

=> LHS = abc/16

=> RHS = a*(b*c)

=> RHS = a*(bc/4)

=> RHS = [ a(bc/4) ] / 4

=> RHS = abc / 4×4

=> RHS = abc/16

.

LHS = RHS ,

Hence , (a*b)*c = a*(b*c) for all a,b,c € Q .