1.1: “And” “Or”

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Bob Dumas and John E. McCarthy
University of Washington and Washington University in St. Louis

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To communicate mathematics you will need to understand and abide by the conventions of mathematicians. In this chapter we review some of these conventions.

Statements are declarative sentences; that is, a statement is a sentence which is true or false. Mathematicians make mathematical statements - sentences about mathematics which are true or false. For instance, the statement:

"All prime numbers, except the number 2, are odd."

is a true statement. The statement: \[" 3<2 . "\] is false.

We use natural language connectives to combine mathematical statements. The connectives "and" and "or" have a particular usage in mathematical prose. Let \(P\) and \(Q\) be mathematical statements. The statement \[P \text { and } Q\] is the statement that both \(P\) and \(Q\) are true.

Mathematicians use what is called the "inclusive or". In everyday usage the statement " \(P\) or \(Q\) " can sometimes mean that exactly one (but not both) of the statements \(P\) and \(Q\) is true. In mathematics, the statement \[P \text { or } Q\] is true when either or both statements are true, i.e. when any of the following hold:

\(P\) is true and \(Q\) is false.

\(P\) is false and \(Q\) is true.

\(P\) is true and \(Q\) is true.