**Prob.** **Let ({a, b}, * ) be a semigroup where a a =b. Show that- ab=b*a.**

## Sol.

Given

({a*b}, *) is a semigroup

And a*a = b.

Now

a*b = a*(a*a) (∵ a*a=b)

a*b =(a*a)*a (by associative law)

a*b =b*a (∵ a*a=b)

**Prob.** **Let ({a, b}, * ) be a semigroup where a a =b. Show that- ab=b*a.**

Given

({a*b}, *) is a semigroup

And a*a = b.

Now

a*b = a*(a*a) (∵ a*a=b)

a*b =(a*a)*a (by associative law)

a*b =b*a (∵ a*a=b)