Question

Use a direct proof to show that the product of two odd integers is odd.

Answer 1

Answer:

The proof itself

Step-by-step explanation:

We can define the set of all even numbers as

$E = \{ a \in \mathbb{Z} \setminus a = 2.k , k\in \mathbb{N}\}$

This is, we can define all even numbers as the set of all the multiples of $2$

As for the odd numbers, we can always take every even number and sum one to each one. This is

$O = \{ a\in \mathbb{Z} \setminus a=2.k+1,k\in\mathbb{N}_{0}\}$

Note that $k\in\mathbb{N}_{0}$(the set of all natural numbers adding the zero) so that for $k=0$ then $a=1$

Now, given 2 odd numbers $a$ and $b$ we can write each one as follows:

$a = 2k+1\\b = 2l+1\\k,l \in\mathbb{N}_{0}$

And then if we multiply them with each other we obtain:

$a.b = (2k+1).(2l+1)\\= 4kl+2k+2l+1\\= 2(2kl+k+l) + 1\\= 2k'+1 \\where\ k'=2kl+k+l$

Then we have that $a.b$ is also an odd number as we defined them.