8 Chapter 1. Introduction
some useful tips. (You may take a look at t he boo k How to Solve I t, by G.
P´olya).
1. Read and completely understand the statement of the theorem to be
proved. Most often this is the hardest part.
2. Sometimes, theorems contain theorems inside them. For example,
“Property A if and only if property B”, requires showing two state-
ments:
(a) If property A is true, then property B is true (A ⇒ B).
(b) If property B is true, then property A is true (B ⇒ A).
Another example is the theorem “Set A equals set B.” To prove this,
we need to prove that A ⊆ B and B ⊆ A. That is, we need to show
that each element of set A is in set B, and tha t each element of set B
is in set A.
3. Try to work out a few simple cases of the theorem just to get a grip on
it (i.e., crack a few simple cases first).
4. Try to write down the proof once you have it. This is to ensure the
correctness of your proof. Often, mistakes are found at the time of
writing.
5. Finding proofs takes time, we do not come pr ewired to produce proofs.
Be patient, think, express and write clearly and try to be precise as
much as possible.
In the next sections, we will go through some of the proof stra tegies.
1.3.1 Direct proofs
As the name suggests, in a direct proof of a theorem, we just approach the
theorem directly.
Theorem 1.3.1 If n is an odd positive integer, then n
2
is odd as well.
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