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Example 0.2.3
Decide which of the following statements are true and which are false. Briefly explain.
$0=1 \rightarrow 1=1$
$1=1 \rightarrow$ most horses have 4 legs
If 8 is a prime number, then the 7624 th digit of $\pi$ is an 8 .
If the 7624 th digit of $\pi$ is an 8 , then $2+2=4$
Solution. All four of the statements are true. Remember, the only way for an implication to be false is for the if part to be true and the then part to be false.
Here the hypothesis is false and the conclusion is true, so the implication is true.
Here both the hypothesis and the conclusion is true, so the implication is true. It does not matter that there is no meaningful connection between the true mathematical fact and the fact about horses.
I have no idea what the 7624th digit of $\pi$ is, but this does not matter. Since the hypothesis is false, the implication is automatically true.
Similarly here, regardless of the truth value of the hypothesis, the conclusion is true, making the implication true.
Example 0.2.4
Prove: If two numbers $a$ and $b$ are even, then their sum $a+b$ is even.
Solution. Suppose the numbers $a$ and $b$ are even. This means that $a=2 k$ and $b=2 j$ for some integers $k$ and $j$. The sum is then $a+b=2 k+2 j=2(k+j)$. Since $k+j$ is an integer, this means that $a+b$ is even.
Notice that since we get to assume the hypothesis of the implication we immediately have a place to start. The proof proceeds essentially by repeatedly asking and answering, “what does that mean?”
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