$(20$ points$)$

$($a$)$ Determine whether the following argument is valid:

$pr$

$qr$

not$(pvvq)$

:$.rr$

$($b$)$ Determine whether the following argument is valid:

$pr$

$qr$

$\frac{qvv-r}{-p}$

Solution:

$3.(20$ points$)$

Consider the following theoren: $"$if $x$ and $y$ are odd integers, then $x+y$ is even."

$($a$)$ Give a direct proof of this theorem.

$($b$)$ Give a proof by contrudiction of this theorem.

Solution: