Question

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

Answer 1

Answer:

Multiplication is commutative.          

Step-by-step explanation:

Let n,m be two natural numbers.

We have to show that $n\times m=m\times n$

We will prove this by induction

For n = 1, we have $n\times 1 = 1\times n = n$, which is true.

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

$(n-1)\times m = m\times (n-1)$

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.