1. ## Contrapositive Method

Plz help solving these:

(a) Assume that $\displaystyle m$ and $\displaystyle n$ are positive integers and that $\displaystyle m \le n$. Prove that if $\displaystyle m^2 = n^2$, then $\displaystyle m=n$ using contrapositive method.

(b) Using direct proof show that the negative of any even integer is even.

2. Originally Posted by cu4mail
Plz help solving these:

(a) Assume that $\displaystyle m$ and $\displaystyle n$ are positive integers and that $\displaystyle m \le n$. Prove that if $\displaystyle m^2 = n^2$, then $\displaystyle m=n$ using contrapositive method.
to prove using the contrapositive method means the following.

by the contrapositive method, we prove a statement $\displaystyle P \implies Q$ by showing $\displaystyle \neg Q \implies \neg P$

so start by assuming $\displaystyle m \ne n$, where does this get you? how does this cause $\displaystyle m^2$ to relate to $\displaystyle n^2$? will they be equal?

(b) Using direct proof show that the negative of any even integer is even.
Hint: we say an integer is even if it can be expressed as $\displaystyle 2n$ for $\displaystyle n \in \mathbb{Z}$. now note that $\displaystyle -2n = 2(-n)$

3. Originally Posted by Jhevon
to prove using the contrapositive method means the following.
by the contrapositive method, we prove a statement $\displaystyle P \implies Q$ by showing $\displaystyle \neg P \implies \neg Q$
Shouldn't that read "by the contrapositive method, we prove a statement $\displaystyle P \implies Q$ by showing $\displaystyle \neg Q \implies \neg P$"

4. Originally Posted by Isomorphism
Shouldn't that read "by the contrapositive method, we prove a statement $\displaystyle P \implies Q$ by showing $\displaystyle \neg Q \implies \neg P$"
yes, indeedy! thanks. i'll fix it (hehe, i must be dyslexic or something, i had to read what you wrote like three times before i saw the difference between what i wrote and what you wrote)

5. Originally Posted by Jhevon
to prove using the contrapositive method means the following.

by the contrapositive method, we prove a statement $\displaystyle P \implies Q$ by showing $\displaystyle \neg Q \implies \neg P$

so start by assuming $\displaystyle m \ne n$, where does this get you? how does this cause $\displaystyle m^2$ to relate to $\displaystyle n^2$? will they be equal?

Hint: we say an integer is even if it can be expressed as $\displaystyle 2n$ for $\displaystyle n \in \mathbb{Z}$. now note that $\displaystyle -2n = 2(-n)$

6. and are positive integers and that . Prove that if , then using contrapositive method.

7. and are positive integers and that . Prove that if , then using contrapositive method.