Question

let * be a binary operation on z defined by a*b=a+b-4, for all a,b&z, find the identity element

Answer 1

a*e=a=e*a
For a binary operation,

If a*e = a then element ‘e’ is known as left identity ,

or If e*a = a then element ‘e’ is known as right identity.

Now,

a*b = a+b-4

So, first check left identity,
a*e=a
a+e-4=a
e=4

Now for left identity,

e*a= a
e+a-4=a
e=4

e’ is both a left identity and a right identity in this case so it is known as two sided identity.

It can be in the form of ‘a’ as long as it belongs to the set on which the operation is defined.

Hence, identity element for this binary operation is e=4