Mathematical Reasoning
The Mathematical Reasoning domain tests logical structure, justification and proof rather than calculation alone. Questions may ask you to evaluate an argument, identify a converse or contrapositive, test a general claim with a counterexample, or choose between inductive and deductive reasoning.
Core rules and concepts
- A statement is a sentence that is either true or false. The negation of ‘all A are B’ is ‘at least one A is not B,’ not ‘no A are B.’
- In an implication P → Q, P is sufficient for Q and Q is necessary for P. The converse is Q → P; the inverse is ¬P → ¬Q; the contrapositive is ¬Q → ¬P.
- An implication and its contrapositive are logically equivalent. The converse and inverse are equivalent to each other but not to the original implication.
- A single counterexample disproves a universal claim. Inductive reasoning generalises from examples and is not a proof; deductive reasoning derives a conclusion from accepted premises.
- Proof by contradiction assumes the negation of the desired conclusion and derives an impossibility.
- In a valid argument, the conclusion must follow from the premises; true premises alone do not make an invalid argument valid.
Worked example
Consider: ‘If a number is divisible by 6, then it is divisible by 2.’ The contrapositive is: ‘If a number is not divisible by 2, then it is not divisible by 6.’ This is logically equivalent and true. The converse, ‘If a number is divisible by 2, then it is divisible by 6,’ is false; 4 is divisible by 2 but not by 6, so 4 is a counterexample. The original statement is not proved by confirming examples such as 12; a general proof is needed.
Traps that cost marks
- Confusing the converse with the contrapositive.
- Disproving a statement with an example that does not satisfy its hypothesis.
- Treating several confirming examples as a proof.
- Assuming that because an argument’s conclusion is true, the argument is valid.
- Negating ‘some’ as ‘some are not’ instead of ‘none.’