In the first chapter we were mainly interested in the meaning of mathematical statements. However, mathematics is primarily concerned with establishing the truth of statements. This is achieved by giving a proof of the statement. The key idea in most proofs is that of implication and this idea is discussed in this chapter.

Implications

A proof is essentially a sequence of statements starting from statements we know to be true and finishing with the statement to be proved. Each statement is true because the earlier statements are true. The justification for such steps usually makes use of the idea of ‘implication’; an implication is the assertion that if one particular statement is true then another particular statement is true.

The symbol usually used to denote implication in pure mathematics is ⇒ although there are a variety of forms of words which convey the same meaning. For the moment we can think of ‘P ⇒ Q’ as asserting that if statement P is true then sq is statement Q, which is often read as ‘P implies Q’. The meaning will be made precise by means of a truth table. Before doing this it is necessary to clarify what this meaning should be and to do this we consider an example concerning an integer n.

Suppose that P(n) is the statement ‘n > 3’ and Q(n) is the statement ‘n > 0’, where n is an integer.