Question

Check the validity of the following statement:
P: if x and y are odd integers, then $\left( x+y \right)$ is an odd integer.
A. true
B. false

Answer 1

Hint: We express the notion of even and odd numbers in their general form. We try to express x and y in that form also. We find the summation of the numbers. Then we find the form of the sum to find whether $\left( x+y \right)$ is an odd integer or not.

Complete step-by-step solution:
We try to express given terms in their general form of odd integers.
Any odd integer can be expressed in the form of $2n+1$ where $n\in \mathbb{Z}$.
Any even integer can be expressed in the form of $2n$ where $n\in \mathbb{Z}$.
We have been given that x and y are odd integers. So, we express them in a general form.
Let x be $x=2k+1$ for some $k\in \mathbb{Z}$ and y be $y=2l+1$ for some $l\in \mathbb{Z}$
We need to find $\left( x+y \right)$. We put their respective values.
So, $\left( x+y \right)=\left[ \left( 2k+1 \right)+\left( 2l+1 \right) \right]$
We find the solution to the equation by simplifying it.
\[\left( x+y \right)=\left[ 2k+1+2l+1 \right]=2\left( k+l \right)+2=2\left( k+l+1 \right)\]
Here, we need to find the characteristics of \[\left( k+l+1 \right)\].
We have assumed that $k\in \mathbb{Z}$, $l\in \mathbb{Z}$ and $1\in \mathbb{Z}$.
We know that the sum of any number of integers will always be integers.
So, \[\left( k+l+1 \right)\] is the sum of three integers. So, the value of \[\left( k+l+1 \right)\] is also integer.
Let’s assume \[\left( k+l+1 \right)=p\in \mathbb{Z}\].
This means $\left( x+y \right)=2\left( k+l+1 \right)=2p$
We already mentioned that any even integer can be expressed in the form of $2n$ where $n\in \mathbb{Z}$.
Here p is an integer. So, $\left( x+y \right)$ is of the form of even number.
Let’s rake example of 19 and 77.
$77+19=96$ which is even.
We can take any number of examples but the results will always be even.
So, the statement is false. The correct option is (B).

Note: We can prove this using the contradiction theorem also where we break an odd number into two odd numbers which are never possible. So, contradiction comes and we get the result. We also need to remember that sum of odd and even numbers will always be odd.