Free lesson · Module 10 · Paper 2

The logic of arguments: if, only if, and the contrapositive

TMUA Paper 2 tests logic that most A-level courses never teach. This lesson covers the vocabulary in the official specification, with the traps examiners use.¹ It takes about 30 minutes.

1. Statements and implications

A statement is something that is either true or false. Most TMUA logic questions are about implications: “if A then B”, also written “A implies B”.

Key idea. “If A then B” is false in exactly one situation: A is true and B is false. If A is false, the statement is automatically true, however strange that sounds.

For example, “if a number is a multiple of 4, then it is even” is true. Some even numbers aren't multiples of 4 (like 6), but that doesn't contradict it, because the statement says nothing about numbers that aren't multiples of 4.

2. Four ways of saying the same thing

These all mean exactly the same as “if A then B”:

A implies B	A only if B
B if A	A is sufficient for B; B is necessary for A

The classic trap: “A only if B” means “if A then B”, not “if B then A”. Read “only if” as “can only happen when”: “you pass only if you revise” means passing requires revising. It doesn't promise that revising guarantees a pass.

“A if and only if B” means both “if A then B” and “if B then A”. Then A and B are always both true or both false.

3. Converse, inverse and contrapositive

Start from “if A then B”. There are three related statements:

Name	Form	Same truth as the original?
Converse	If B then A	Not necessarily
Inverse	If not A then not B	Not necessarily
Contrapositive	If not B then not A	Always

Worked example. Original: “If a quadrilateral is a square, then it has four equal sides.” True.
Converse: “If it has four equal sides, then it is a square.” False: a rhombus is a counterexample.
Contrapositive: “If it doesn't have four equal sides, then it isn't a square.” True, as it must be.

The contrapositive is a powerful proof tool. To prove “if n² is even then n is even”, it is easier to prove the contrapositive: “if n is odd then n² is odd”. If n = 2k + 1, then n² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is odd.

4. Necessary and sufficient

“A is sufficient for B” means A is enough to guarantee B (if A then B). “A is necessary for B” means B can't happen without A (if B then A).

Statement about real x	Verdict
x = 3 is ___ for x² = 9	Sufficient, not necessary (x = −3 also works)
x > 0 is ___ for x > 5	Necessary, not sufficient (x = 1)
x³ = 8 is ___ for x = 2	Necessary and sufficient (cubing is one-to-one)
x is an integer is ___ for x > 1	Neither (x = 0 and x = 1.5)

Exam method. For “A is ___ for B”, test two things separately. Does A force B? If yes, A is sufficient. Does B force A? If yes, A is necessary. Disprove either with one counterexample.

5. “For all”, “there exists”, and negation

“For all x, P(x)” is disproved by a single counterexample. “There exists x with P(x)” is proved by a single example. In the TMUA, “some” means “at least one”.¹

To negate a statement, swap each quantifier and negate the condition at the end:

Statement	Negation
For all x, P(x)	There exists x with not P(x)
There exists x with P(x)	For all x, not P(x)
A and B	not A or not B
A or B	not A and not B
If A then B	A and not B

Trap: the negation of “if A then B” is not “if A then not B”. It is “A is true and B is false”: the one situation that breaks the implication.

Check yourself

Write the contrapositive of “if a triangle is right-angled, then a² + b² = c² for some labelling of its sides”.
Is “n is divisible by 6” necessary, sufficient, both or neither for “n is divisible by 3”?
Negate: “Every student in the class has a sibling who is older than them.”

Show answers

If a² + b² ≠ c² for every labelling of its sides, then the triangle is not right-angled.
Sufficient but not necessary: 6 | n forces 3 | n, but n = 3 is divisible by 3 and not by 6.
There is a student in the class who has no older sibling.

Test yourself: questions 6–10 are Paper 2 style

Sources

TMUA content specification (Section 2: Arg1–Arg4, Prf1–Prf5, Err1–Err2), UAT-UK