Prove if m + n ≥ 59 then (m ≥ 30 or n ≥ 30) by Contraposition

Question

I need help with this contraposition.

Question Let m, n ≥ 0 be integers. Prove that if m + n ≥ 59 then (m ≥ 30 or n ≥ 30).

What I've done so far

I'm not exactly sure if I am setting this up right. Since my I am using proof by contraposition I am setting it up as.

m, n ≤ 0 If m + n ≤ 59 then (m ≤ 30 or n ≤ 30).

I don't know where to go from here. Any help would be appreciated thanks!

Try a proof by contradiction. That means assume that the statement you want to prove is false, then work logically and correctly to show that some false statement is made true. — Faraz Masroor, Sep 27 '15 at 00:53
@FarazMasroor The question requires a proof by contraposition, not contradiction - there is a technical difference: http://math.stackexchange.com/questions/262828/proof-by-contradiction-vs-prove-the-contrapositive — Deepak, Sep 27 '15 at 00:59

score 1 · Answer 1 · answered Sep 27 '15 at 00:54

1

That's not the contrapositive.

Instead try:

IF $(m<30)$ AND $(n<30)$ THEN $(m+n < 59)$.

(Remember you're dealing with integers here).

answered Sep 27 '15 at 00:54

Deepak

score 0 · Answer 2 · answered Sep 27 '15 at 00:55

If $p\implies q$ then the contrapositive of it is $\neg\ q\implies \neg p$ and those two statements are equivalent.

Applying it to this case, we have to prove that if $m<30 \land n<30$ then $m+n<59.$

However, this is obvious so we have proved the first statement by contraposition.

QED.

2 Answers2