NSW HSC Mathematics Extension 2

## Statements A **statement**, also called a **proposition**, is a sentence with a definite truth value: it is true or it is false, and never both. "$7$ is prime" is a statement (true), "$2 + 2 = 5$" is a statement (false), and "$x > 3$" is not a statement on its own, because its truth depends on the value of $x$. It becomes a statement once $x$ is given a value or is quantified, as in "for all real $x$, $x^2 \ge 0$". Statements are usually named with capital letters such as $P$ and $Q$, and new statements are built from them with connectives. ## Connectives The **conjunction** $P \land Q$ is read "$P$ and $Q$" and is true only when both $P$ and $Q$ are true. The **disjunction** $P \lor Q$ is read "$P$ or $Q$" and is true when at least one of them is true, so it includes the case where both are true. The **negation** of $P$ is "not $P$", written $\neg P$ or $\sim P$. It is true exactly when $P$ is false, and negating twice returns the original statement: $$ \neg(\neg P) = P. $$ The **implication** "if $P$ then $Q$" is written $P \Rightarrow Q$ or $P \to Q$ and read "$P$ implies $Q$". The statement $P$ is the hypothesis and $Q$ the conclusion. An implication is false in exactly one situation: $P$ true and $Q$ false. When $P$ is false the implication is true whatever $Q$ is, because nothing has been promised. For example, the statement "if $x = 3$ then $x^2 = 10$" is true when it is applied to $x = 5$, because its hypothesis is false there and nothing is claimed. It is false…