Year 11 / Unit 1
Make every implication count
A proof establishes a claim for every allowed case. A counterexample disproves a universal statement. Testing many cases can suggest a claim but cannot establish it.
In a direct proof, begin with definitions and deduce the conclusion. Contraposition proves “not Q implies not P” instead of “P implies Q”. Contradiction assumes the claim fails and derives an impossibility.
Prove that the sum of two odd integers is even.
- Write the integers as 2a + 1 and 2b + 1, with a and b integers.
- Their sum is 2(a + b + 1).
- The bracket is an integer, so the sum is even.
Assumed knowledge
Use definitions of integers, even and odd numbers; expand and factorise expressions; and follow deductive arguments from elementary geometry. Distinguish a statement about every case from an example.
Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.
QCAA prior-learning guidance ↗