Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

Beginning Activity Beginning Activity 1: Compound Statements

A4US

Mathematicians often develop ways to construct new mathematical objects from existing mathematical objects. It is possible to form new statements from existing statements by connecting the statements with words such as “and” and “or” or by negating the statement. A logical operator (or connective) on mathematical statements is a word or combination of words that combines one or more mathematical statements to make a new mathematical statement. A compound statement is a statement that contains one or more operators. Because some operators are used so frequently in logic and mathematics, we give them names and use special symbols to represent them.

1. The conjunction of the statements $P$ and $Q$ is the statement “$P$ and $Q$” and its denoted by $P\wedge Q$ . The statement $P\wedge Q$ is true only when both $P$ and $Q$ are true.

2. The disjunction of the statements $P$ and $Q$ is the statement “$P$ or $Q$” and its denoted by $P\vee Q\text{.}$ The statement $P\vee Q$ is true only when at least one of $P$ or $Q$ is true.

3. The negation of the statement $P$ is the statement “not $P$” and is denoted by $\mathrm{\neg }P$ . The negation of $P$ is true only when $P$ is false, and $\mathrm{\neg }P$ is false only when $P$ is true.

4. The implication or conditional is the statement “If $P$ then $Q$” and is denoted by $P\to Q$ . The statement $P\to Q$ is often read as “$P$ implies $Q\text{,}$” and we have seen in Section 1.1 that $P\to Q$ is false only when $P$ is true and $Q$ is false.