epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Special Kinds of Theorems
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Diagram 2
Diagram 3
Diagram 4
Depending on our priorities, we can choose the chain of proofs that involves
the imphcations that are easier to prove, or the one that gives the more
detailed information, or the one that requires the smallest number of proofs.
Therefore, there is no prescribed way of proving that three or more
statements are equivalent.
In general, to prove that n statements are equivalent, one needs to prove
at least n imphcations (see Diagram 1 for an illustration of this claim).
Let us now consider some more examples.
EXAMPLE 3. Let a and b be two distinct real numbers. Then the following
statements are equivalent:
i. The number b is larger than the number a.
ii. Their average, (a + b)/2, is larger than a,
iii. Their average, (a + ft)/2, is smaller than b.