To some extent we have been generalizing and specializing throughout the book. Discussing it in detail has been delayed until now as one needs to see examples before really getting to grips with these ideas.

Generalization

Weakening the hypotheses

Given a theorem a question you should ask is ‘Can I weaken the hypothesis and still get the same conclusion?’ The objective of mathematics is to use as few hypotheses as possible to get as strong a conclusion as possible.

That is, use as little as possible to say as much as possible. The statement given by weakening the assumptions is called a generalization.

Let us consider the simple theorem on even numbers:

‘If x and y are even natural numbers, then x + y is even.’

We can weaken the hypothesis to the statement

‘If x and y are even integers, then x + y is even.’

This statement is true.

We could have weakened the hypothesis in a different direction by dropping the requirement that the numbers are even:

‘If x and y are natural numbers, then x + y is even.’

However, note that this statement is false; take x = 2 and y = 3 for instance. Thus weakening (or, as in this case, losing) an assumption in a theorem can lead to a false statement.