Content Specification

SECTION 2

This section sets out the scope of Paper 2. Paper 2 tests the candidate’s ability to think mathematically: the paper will focus on testing the candidate’s ability to understand, and construct, mathematical arguments in a variety of contexts. It will draw on the mathematical knowledge outlined in SECTION 1 in the test specification.

The Logic of Arguments

Arg1 Understand and be able to use mathematical logic in simple situations:

• The terms true and false;

• The terms and, or (meaning inclusive or), not;

• Statements of the form:

  if A then B
  A if B
  A only if B
  A if and only if B

• The converse of a statement;

• The contrapositive of a statement;

• The relationship between the truth of a statement and its converse and its contrapositive.

Note: candidates will not be expected to recognise or use symbolic notation for any of these terms, nor will they be expected to complete formal truth tables.

Arg2 Understand and use the terms necessary and sufficient.

Arg3 Understand and use the terms for all, for some (meaning for at least one), and there exists.

Arg4 Be able to negate statements that use any of the above terms.

Mathematical Proof

Prf1 Follow a proof of the following types, and in simple cases know how to construct such a proof:

• Direct deductive proof (‘Since A, therefore B, therefore C, …, therefore Z, which is what we wanted to prove.’);
• Proof by cases (for example, by considering even and odd cases separately);
• Proof by contradiction;
• Disproof by counterexample.

Prf2 Deduce implications from given statements.

Prf3 Make conjectures based on small cases, and then justify these conjectures.

Prf4 Rearrange a sequence of statements into the correct order to give a proof for a statement.

Prf5 Problems requiring a sophisticated chain of reasoning to solve.

Identifying Errors in Proofs

Err1 Identifying errors in purported proofs.

Err2 Be aware of common mathematical errors in purported proofs; for example, claiming ‘if $ab = ac$, then $b = c$’ or assuming ‘if $\sin A = \sin B$, then $A = B$’ neither of which are valid deductions.