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38 The Nuts and Bolts of Proof, Third Edition
implications will have to be proved explicitly, while others might follow from
some of the impHcations already proved.
Let us assume that we want to prove that four statements, A, B, C, and D,
are equivalent. There are many ways of proceeding. We will look at four of
them, working in detail on the first one and just giving the outlines
(diagrams) for the others. It will be up to you to check that by proving the
implications represented by the arrows in each diagram, we would indeed
prove that the four statements A, B, C, and D are equivalent.
Diagram 1
We use this diagram to express the fact that we have proved the following
implications:
i. If A, then B.
ii. If B, then C.
iii. If C, then D.
iv. If D, then A.
The order in which the implications have been proved is not relevant.
We can see that, if these four implications are true, then A implies B, C,
and D. Indeed:
•
a. A implies B (proved explicitly).
b. A implies B, and B impHes C; so A imphes C.
c. A implies C, and C implies D; so A implies D.
On the other hand, A is impUed by the other three statements:
a. D implies A (proved explicitly).
b. C impHes D, and D implies A; so C implies A.
c. B implies C, and C imphes A; so B implies A.
Similarly, we can establish that B, C, and D imply all other statements and
are implied by all of them.
C