Question

Lemma 2.3.2 (Multiplication is commutative). Let n,m be natural питbers. hen n x m = mхп.

Answer 1

Multiplication is commutative.

Explanation:

Let n,m be two natural numbers.

We have to show that
`n* m=m* n`

We will prove this by induction

For n = 1, we have
`n* 1 = 1* n = n`, which is true.

Let the given statement be true for n-1 natural numbers, that is,

`(n-1)* m = m* (n-1)`

Now, we have to show that it is true for n natural numbers.

`nm = (n-1)m + m = m(n-1) + m`

which is equal to

`(m-1)(n-1) + (n-1) + m`

`= (m-1)(n-1) + (m-1) + n`

`= n(m-1) + n`

`= (m-1)n + n`

`= mn`

Hence, by principal of mathematical induction, the given statement is true for all natural numbers.