Skip to main content Accessibility help
Internet Explorer 11 is being discontinued by Microsoft in August 2021. If you have difficulties viewing the site on Internet Explorer 11 we recommend using a different browser such as Microsoft Edge, Google Chrome, Apple Safari or Mozilla Firefox.

Chapter 2: Implications

Chapter 2: Implications

pp. 10-20

Authors

, University of Manchester
  • Add bookmark
  • Cite
  • Share

Summary

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 ‘PQ’ 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.

About the book

Access options

Review the options below to login to check your access.

Purchase options

Hardback
US$195.00
Paperback
US$54.00

Have an access code?

To redeem an access code, please log in with your personal login.

If you believe you should have access to this content, please contact your institutional librarian or consult our FAQ page for further information about accessing our content.

Also available to purchase from these educational ebook suppliers