supply a proof for the assertion 15 The integer zv is even i

supply a proof for the assertion

15. The integer z-v is even if and only if at one of the integers r and y least is even.

Solution

Let x,y be integers.

(xy is even) (x is even or y is even)

The contrapositive is then ¬(x is even or yy is even)¬(xy is even).

This means we want to prove that if x is odd AND y is odd, then xy is odd.

Start in the standard way: Let x=2a+1 and let y=2b+1 where a,bZ.

Then calculate xy, and represent the product as an odd integer.