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``mcs'' --2015/5/18 --1:43 -- page 13 --\# 21
{\it 1.6 Proving an} ``{\it If and Only If}'' {\it 13}
1.5.2 Method \# 2- Prove the Contrapositive
An implication $(P$ IMPLIES $Q$ is logically equivalent to its {\it contrapositive}
NOT ({\it Q}) IMPLIES NOT ({\it P}) .
Proving one is as good as proving the other, and proving the contrapositive is some- times easier than proving the original statement. If so, then you can proceed as follows:
1. Write, ``We prove the contrapositive:'' and then state the contrapositive.
2. Proceed as in Method \# 1.
Example
Theorem 1.5.2. {\it If} $r$ {\it is irrational, then} $\sqrt{r}$ {\it is also irrational}.
A number is {\it rational} when it equals a quotient of integers --that is, if it equals $m/n$ for some integers $m$ and $n$. If it's not rational, then it's called {\it irrational}. So we must show that if $r$ is {\it not} a ratio of integers, then $\sqrt{r}$ is also {\it not} a ratio of integers. That's pretty convoluted! We can eliminate both {\it not}'{\it s} and simplify the proof by using the contrapositive instead.
{\it Proo}. We prove the contrapositive: if $\sqrt{r}$ is rational, then $r$ is rational.
Assume that $\sqrt{r}$ is rational. Then there exist integers $m$ and $n$ such that:
$$
\sqrt{r}=\frac{m}{n}
$$
Squaring both sides gives:
$$
r\ =\frac{m^{2}}{n^{2}}
$$
Since $m^{2}$ and $n^{2}$ are integers, $r$ is also rational. $\blacksquare$
1.6 Proving an ``If and Only If''
Many mathematical theorems assert that two statements are logically equivalent; that is, one holds if and only if the other does. Here is an example that has been known for several thousand years:
Two triangles have the same side lengths if and only if two side lengths
and the angle between those sides are the same.
The phrase ``if and only if'' comes up so often that it is often abbreviated ``iff.''
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