VCE Specialist Mathematics

## Statements and conjectures A **statement** (or proposition) is a sentence that is either true or false, never both. "$7$ is prime" and "every square number is even" are statements, the first true and the second false. "$x + 2 = 5$" is not a statement on its own, because its truth depends on $x$; it becomes one when $x$ is given a value or quantified. A **conjecture** is a statement put forward as possibly true, which has then to be proved or disproved. Mathematicians usually reach one by looking at examples and spotting a pattern. The values $n^2 - n + 41$ are prime for $n = 1, 2, 3, \dots, 40$, which might suggest that the expression is always prime, but $n = 41$ gives $41^2$, which is not. Examples therefore support a conjecture without establishing it, whereas one failure disproves it. ## Connectives Statements are combined with the connectives "and" ($\wedge$), "or" ($\vee$) and "not" ($\neg$). In mathematics "or" is inclusive: $P \vee Q$ is true when $P$ is true, when $Q$ is true, or when both are. | $P$ | $Q$ | $P \wedge Q$ | $P \vee Q$ | $\neg P$ | |---|---|---|---|---| | T | T | T | T | F | | T | F | F | T | F | | F | T | F | T | T | | F | F | F | F | T | The negation of "$P$ and $Q$" is "not $P$ or not $Q$", and the negation of "$P$ or $Q$" is "not $P$ and not $Q$". For example, the negation of "$x > 2$ and $x < 5$" is "$x \le 2$ or $x \ge 5$". ## Implications The **implication** $P \Rightarrow Q$ ("if $P$ then $Q$", "$P$ implies $Q$") says that whenever $P$ is…