Show (by giving a step-by-step algebraic derivation from the LHS to the RHS) that if 〈?,∗〉 is a binary algebraic structure where ∗ is commutative, then for all a, b, c ϵ S a * (b*c) = (c*b) * a

Step-by-step explanation:

From the left hand side,

Let b*c = P, and c*b = P'

Then a * (b*c) = a*P

Because * is commutative,

b*c = c*b

a*P = P*a

a*P' = P' * a

are all true

So P = P'

a*P = a*P' (Since P = P')

a*P = P' * a (Since a*P' = P' * a)

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

For all a, b, c in S