A proof by contradiction is when we are proving a statement where
we are assuming the hypothesis is true and the conclusion is false.
For instance, we can say that 2–√ is irrational. For contradiction we
would assume that 2–√ is rational so we can say that 2–
√=pq where p and q do not = 0 and are integers with no common
factors. We would start by squaring and multiplying by q2 to
get p2=2(q2) this means that p2or p is an even number. We
can say that p=2k for some an integer k. So now we can say
that 4k2=2q2 which means that q2=2k2 which means that
q and q2 are even and because they are both even that means they
are both divisible by 2. This contradicts what was said about p and q
having no common factors. This means that we cannot have 2–√ as
a rational number which means it is rational. A proof by
contrapositive is basically saying that if one statement is true than its
opposite or negation must also be true or its equivalent.
