Check whether the following statement is true or not. If x, `y in Z`are such that x and y are odd, then xy is odd.

Explanation

Let,P, `x,y in Z` such that x and y are odd
And q: xy is odd
Then , we have to prove that xy is odd
DIERECT METHOD
We assume that p is the and show that q is true
P is true means x and y are odd interger m
and y= (2m +1) for some interger m M
`thererfore` xy =(2m+1)(2n+1)
`=(4 mn+2m+2n+1)`
`=2(2mn+m+nb) +1 ` which is clearly odd.
Thus `p rArr q.`
Hence , the fiven statement is true.
CONTRAPSITIVE METHOD
We will shoe that `~q rArr ~ p`
Here ~ p It is false that the both x and y are odd .
This means at least one of x and y is even
Let x be even . Then x= 2n for some interger n
`therefore` xy = 2ny for some that xy is even.
Thus ~ p is true `[ because ~ P : ` xy is even ]
`therefore ~ Q rArr ~ P`
So the given statement is ture.