Let P (x,y) denote the sentence 2x+y=5. what are truth value of the following domain of x and y is the set of all integers?

Answer is

{(x,y):2x+y=5,x,y∈Z}=(2−t,2t+1)

 Proof.

Firstly let us assume that 2x+y=5 with x,y∈Z.

We have y=5-2x is odd as difference odd value 5 and even value 2x.

Each odd number can be written in the form y=$2\cdot t+1,\space where\\$

$t=\frac{y-1}{2}\in Z\space because\space y-1\space is\space even,$

Therefore x=$\frac{5-y}{2}=\frac{5-2t-1}{2}=\frac{4-2t}{2}=2-t$ .

So we have (x,y)=$(2-t,\space 2t+1),\space where\space t\in Z.$

Conversly, if (x,y)=$(2-t,\space 2t+1),$

then we have that y is odd and 2⋅(2−t)+2t+1=5 .

The domain of the equation is not the set of all integers.