Paper 2 lives here. 'If \(A\) then \(B\)' has converse 'if \(B\) then \(A\)' (not equivalent) and contrapositive 'if not \(B\) then not \(A\)' (equivalent). '\(A\) only if \(B\)' means \(A \Rightarrow B\); '\(A\) if \(B\)' means \(B \Rightarrow A\). \(P\) is sufficient for \(Q\) when \(P \Rightarrow Q\) and necessary when \(Q \Rightarrow P\). Negate quantifiers by swapping 'for all' and 'there exists' and negating the inside.
Contents: Implications (Arg1, Arg2)Quantifiers and negation (Arg3, Arg4)Must-knowTrapsReal questions
Implications (Arg1, Arg2)
- Statement \(A \Rightarrow B\). Converse \(B \Rightarrow A\). Inverse (not \(A\)) \(\Rightarrow\) (not \(B\)). Contrapositive (not \(B\)) \(\Rightarrow\) (not \(A\)). Statement \(\equiv\) contrapositive; converse \(\equiv\) inverse; the two pairs are independent of each other.
- '\(A\) if and only if \(B\)' = both directions.
- \(P\) sufficient for \(Q\): \(P \Rightarrow Q\). \(P\) necessary for \(Q\): \(Q \Rightarrow P\) (you cannot have \(Q\) without \(P\)). Decide each direction separately and look for a counterexample to the one that fails.
- \(A \Rightarrow B\) is equivalent to '(not \(A\)) or \(B\)'.
- Truth of a conditional: 'if \(x > 2\) then \(x^2 > 4\)' is judged for all \(x\); one counterexample makes it false.
Quantifiers and negation (Arg3, Arg4)
- 'For all \(x\), \(P(x)\)' is refuted by one counterexample; 'there exists \(x\) with \(P(x)\)' is proved by one example.
- Negations: not(for all \(x\), \(P\)) = there exists \(x\) with not \(P\); not(exists \(x\) with \(P\)) = for all \(x\), not \(P\); not(\(A\) and \(B\)) = (not \(A\)) or (not \(B\)); not(\(A\) or \(B\)) = (not \(A\)) and (not \(B\)); not(\(x > 3\)) is \(x \leq 3\).
- Order of quantifiers matters: 'for every \(x\) there is \(y > x\)' is true; 'there is \(y\) greater than every \(x\)' is false.
- Nested negation: 'for every \(n\) there is a prime \(p\) with \(n < p < 2n\)' negates to 'there is an \(n\) for which no prime \(p\) satisfies \(n < p < 2n\)'.
Must-know (no formula booklet)
- contrapositive \(\equiv\) original
- converse \(\not\equiv\) original
- \(P\) sufficient: \(P \Rightarrow Q\); \(P\) necessary: \(Q \Rightarrow P\)
- \(A\) only if \(B\): \(A \Rightarrow B\)
- not(for all) = exists not
Classic traps
- '\(P\) is necessary and sufficient' needs both implications; check for a counterexample in each direction (e.g. \(4 \mid n\) vs \(8 \mid n^2\) really is both).
- 'Not more than 50' is '50 or less', not 'less than 50'.
- A true statement can have a false converse (equilateral \(\Rightarrow\) isosceles).
Worked example
Q. Negate 'every student scored more than 50'.
A. 'At least one student scored 50 or less.'
How it has been examined
2023 · Paper 2Question 7necessary but not sufficient: line with positive gradient and interceptquestion papersit online
2022 · Paper 2Question 4sufficient information to find a distance to a circle centrequestion papersit online
2022 · Paper 2Question 5line through (1, 2): statement, converse, contrapositivequestion papersit online