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Summary
- 1.
- Look up the definitions of any terms and
symbols about which you are doubtful.
2.- Identify the hypotheses, write them down, and
write down any reformulations which appear
useful.
3.- Identify the conclusions to be proved, write
them down, separating them into various parts;
write down any reformulations which appear
promising.
The main things to be done here are:
- (i)
- Look up the definition of any terms (words or
symbols) about which you are uncertain. Likely places
to look are your lecture notes, and textbooks; the
index of a textbook usually includes a reference to
each definition; some textbooks have a separate index
of symbols. By the time that you have done this, you
should be able to translate the statement into
ordinary English.
- (ii)
- Decide which parts of the statement are
hypotheses which you are supposed to assume. Write
them down (``We assume that ...''); include any
reformulations of the hypotheses which look as though
they might be useful; for example, you may replace a
technical mathematical term by its definition.
- (iii)
- Identify the conclusions which you are supposed
to prove; write them down (``We have to prove that
...''). If there is more than one part to the
conclusion, write them down separately. If
appropriate, write down any reformulations of the
conclusion which appear promising; for example, it
may be useful to put the conclusion in a more
mathematical form, by replacing technical terms by
their definitions.
In the context of
Example 0. Let A, B, and C be sets. Prove
that (A ÈB) Ç(AÈC) = A
È(B
ÇC).
We see:
Definitions: It is likely that the reader is
familiar with the notions of union and intersection,
and the symbols È,
Ç which denote
them.
Hypotheses: The only hypothesis is that A, B
and C are sets. (It is implicit that they are all
considered to be subsets of some universal set.)
Conclusions: We have to prove that (A ÈB)
Ç(A ÈC)
= A È(B ÇC). To do this, we should
show:
- (a)
- If x Î (A ÈB)
Ç(A
ÈC), then x
Î A
È(B
ÇC);
- (b)
- If x Î A È(B
ÇC), then x
Î (A
ÈB)
Ç(A
ÈC).
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Introduction
