Last Update: 22 October 2010
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try to show A_{1} → B and … and try to show A_{n} → B
You can prove this semantically with a quick t-table to convince yourself of this
or you can prove it syntactically (i.e., formally).
So, to show (n is an integer) →
(n^{2} ≥ n),
we only have to show:
and this has the form of a proof by cases.
Case (c) is a nice example of a tricky proof
where it's more important for you to understand the proof
than it is to worry about how you would have come up with
the trick at the end of it.
QED
try one of these kinds of proofs:
Or: Show that one of two (or more) candidates for x
satisfies P,
but we just don't know which one it is (see IIB):
i.e., there are 2 irrational numbers such that when one is raised to the power of the other, the result is rational (!!)
proof:
WE DON'T KNOW!
But one of them works, and that's all that we were asked to prove!
QED
Not all mathematicians or computer scientists accept non-constructive proofs.
By the way, although our non-constructive proof doesn't show which one is rational, the answer is given here :-)
For a summary, see Proof Strategies
NO!
but, fairly recently, it was proved.