Let mn N Prove that if m n 100 then either m 50 or n 50

Let m,n N. Prove that if m + n >= 100, then either m >= 50 or n >= 50. What type of proof?

Solution

Let

p be the statement m+n>=100

q be the statement m >= 50

r be the statement n >= 50. We need to prove \"p implies (q or r)\". It is enough to prove its contra positive

\" (q or r)\' implies p\' \"

(q or r)\'= m < 50 and n< 50

p\'=(m+n)<100.

but (m < 50 and n< 50 ) implies (m+n)<50+50=100, which is p\', hence the result.