Mumbai University > Computer Engineering > Sem 3 > Discrete Structures

Marks: 6 Marks

Year: Dec 2015

Let the given statement be P (n). Then,

P(n): (ab)n = anbn.

When = 1, LHS = (ab)1 = ab and RHS = a1b1 = ab

Therefore LHS = RHS.

Thus, the given statement is true for n = 1, i.e., P(1) is true.

Let P(k) be true. Then,

P(k): (ab)k = akbk.

Now, (ab)k + 1 = (ab)k (ab)

= (akbk)(ab) [using (i)]

= (ak ∙ a)(bk ∙ b) [by commutativity and associativity of multiplication on real numbers]

= (ak + 1 ∙ bk + 1 ).

Therefore P(k+1): (ab)k + 1 = ((ak + 1 ∙ bk + 1)

⇒ P(k + 1) is true, whenever P(k) is true.

Thus, P(1) is true and P(k + 1) is true, whenever P(k) is true.

Hence, by the principle of mathematical induction, P(n) is true for all n ∈ N.