Logic

Learn to distinguish mathematical statements (sentences that are definitively true or false) from non-statements such as questions, commands, and opinions, and recognise open sentences whose truth value depends on a variable.

Learn how mathematical "and" requires both parts to hold and mathematical "or" requires at least one part to hold, including why "or" in mathematics is always inclusive.

Learn how negation flips truth values and how to negate compound "and" and "or" statements using De Morgan's laws.

Learn to read conditional statements of the form "if $P$ then $Q$", identify the hypothesis and conclusion, and understand that a conditional makes no claim when its hypothesis is false.

Learn how to form the converse of a conditional by swapping hypothesis and conclusion, and determine whether the converse is true or false using counterexamples.

Learn how to form the contrapositive of a conditional by negating both parts and swapping them, and use the fact that a conditional and its contrapositive always share a truth value.

Learn how to translate between conditional statements and the language of necessary and sufficient conditions, and classify conditions as necessary, sufficient, both, or neither.

Understand biconditional statements (if and only if), recognise that definitions are always biconditionals, and distinguish definitions from theorems that only guarantee one direction.

Learn to read and interpret universal statements ("for all") and existential statements ("there exists"), and recognise that mathematical conditionals containing variables are universal statements in disguise.

Learn how negating a universal statement ("for all") produces an existential statement ("there exists") and vice versa, and apply these rules to negate mathematical claims.

Learn why the only way a conditional "if $P$ then $Q$" can be false is when $P$ is true and $Q$ is false, and use this to find counterexamples to conditional claims involving variables.

Learn how to apply three fundamental inference rules - modus ponens, modus tollens, and elimination - to draw valid conclusions from given premises, and recognise common fallacies like the converse error.