If and only if
If and only if statements
A theorem that asserts that two statements \(P\) and \(Q\) are equivalent requires you to prove both \(P\implies Q\) and \(Q\implies P\).
Chapter 7 exercise 3.
Proposition: If \(a\) is an integer, then \(a\) is even if and only if \(a^3+a^2+a\) is even.
There are two claims:
\(a\) even implies \(a^3+a^2+a\) is even.
\(a^3+a^2+a\) is even implies \(a\) is even.
Each requires proof.
Proof: First we show that, if \(a\) is even, then \(a^3+a^2+a\) is even.
Now we show that, if \(a^3+a^2+a\) is even, then \(a\) is even.
Chapter 7, exercise 9.
Proposition: Suppose that \(a\) is an integer. Then \(14|a\) if and only if \(7|a\) and \(2|a\).
Proof: First we suppose that \(14|a\) and show that both \(7\) and \(2\) divide \(a\).
Now we show that, if both \(7\) and \(2\) divide \(a\), then \(14\) divides \(a\).
Equivalence
Cycle proofs
If each step in the circle of implications:
\[P_1\implies P_2\implies P_3\implies\cdots\implies P_{n}\implies P_1\]
is true, then all of the statements are equivalent – that is, all true or all false together.