TMUA Paper 2 · Mathematical Reasoning · 2026

Paper 2 logic, decoded.

Necessary vs sufficient, conditionals, contrapositive, quantifiers and counterexamples. The logic language A-level never teaches you, in one page, with TMUA-style questions to check it stuck.

Direction is the mark

If, if, only if

Most Paper 2 errors are direction errors. The two clauses in a conditional are usually simple; the hard part is deciding which clause starts the implication. Translate the sentence into arrow form before judging whether it is true. Do not decide from how familiar the words feel.

English	Shorthand	Read it as
If A, then B	A ⇒ B	A guarantees B.
B if A	A ⇒ B	The word "if" tags the condition.
A only if B	A ⇒ B	B is required for A.
A if and only if B	A ⇔ B	Each one guarantees the other.

“Only if” is not emphasis

“n is even if n is divisible by 4” is true. “n is even only if n is divisible by 4” is false: it claims every even number is divisible by 4, and n = 6 kills it. Same clauses, opposite arrows.

Test yourself 1Conditional direction

Which statement is equivalent to "x is an integer only if 2x is an integer"?

If 2x is an integer, then x is an integer.
If x is an integer, then 2x is an integer.
x is an integer if and only if 2x is an integer.
x is not an integer if 2x is not an integer.
2x is an integer only if x is an integer.

Reveal answer

"A only if B" means A ⇒ B. Here A is "x is an integer" and B is "2x is an integer", so the claim is: if x is an integer, then 2x is an integer.

Two labels for one arrow

Necessary and sufficient

Necessary and sufficient are not extra topics. They label the two ends of a single implication. If A ⇒ B, then A is sufficient for B, and B is necessary for A. If A happens, B must happen: A is enough; B is required. Say the direction aloud until it is automatic.

Sufficient condition

Necessary condition

English	Shorthand	Example
A is sufficient for B	A ⇒ B	Divisible by 4 is sufficient for even.
B is necessary for A	A ⇒ B	Being even is necessary for being divisible by 4.
A is necessary and sufficient for B	A ⇔ B	Both directions hold.

Test yourself 2Necessary / sufficient

Let n be an integer. Which statement is true?

Being divisible by 6 is necessary for being divisible by 3.
Being divisible by 3 is sufficient for being divisible by 6.
Being divisible by 6 is sufficient for being divisible by 3.
Being prime is necessary for being odd.
Being odd is sufficient for being prime.

Reveal answer

If 6 divides n, then 3 divides n, so divisibility by 6 is sufficient for divisibility by 3. A and B fail for n = 3 (divisible by 3, not by 6). D and E both fail for n = 9, which is odd but not prime.

Do not swap without negating

Converse and contrapositive

The converse of a true statement can be false. The contrapositive always has the same truth value as the original. For A ⇒ B, the safe equivalent is “if not B, then not A”.

Name	Shorthand	Meaning
Original	A ⇒ B	If A, then B.
Converse	B ⇒ A	Not automatically equivalent.
Inverse	not A ⇒ not B	Not automatically equivalent.
Contrapositive	not B ⇒ not A	Equivalent to the original.

Use a concrete number

Original: if n is divisible by 4, then n is even. True.
Converse: if n is even, then n is divisible by 4. False; n = 6.
Contrapositive: if n is not even, then n is not divisible by 4. True.

The common error

Proving B ⇒ A when the question asks for A ⇒ B proves the converse, not the original. A proof in the wrong direction earns nothing even if every algebraic line is correct. Paper 2 builds wrong options out of exactly this.

All means every. Some means at least one.

Quantifiers and negation

Before doing any algebra, decide whether the statement needs every case, one case, or no cases. “For some integers n” means at least one integer; it does not mean many, typical, or more than half. A negation is not a stronger claim that sounds opposite: it is the weakest claim that makes the original false. Work from the outer structure inward.

Original	Negation
All objects have property P.	At least one object does not have P.
There exists an object with property P.	No object has property P.
P and Q	not P or not Q
P or Q	not P and not Q
If P, then Q	P and not Q
x < a	x ≥ a

The trap negation

“Every student scored above 70” is not negated by “no student scored above 70”. That is far stronger than needed. The exact negation is “at least one student scored 70 or below”. One exception is enough.

Test yourself 3Quantifier negation

Which is the negation of "For every integer n, n² + n is even"?

For every integer n, n² + n is odd.
There exists an integer n such that n² + n is odd.
There exists an integer n such that n² + n is even.
No integer n has n² + n even.
For every integer n, n² + n is not odd.

Reveal answer

The original is universal, so its negation is existential. Since "not even" means odd for integers, the core property becomes odd. Options A and D are the trap negations: both are far stronger claims than the exact opposite.

One good example beats a long argument

Counterexamples

A counterexample is a permitted case where the assumptions hold but the conclusion fails. Both halves, explicitly checked. An example that breaks an assumption disproves nothing, however dramatic it looks. Search in this order:

Check the universe: integers, positive integers, real numbers, triangles, functions.
Keep the assumptions true: a counterexample that breaks an assumption is irrelevant.
Try small cases: 0, 1, 2, −1, ½.
Try boundary cases: equality cases, tangency, repeated roots, endpoints.

Counterexample anatomy

Claim: if a² = b² then a = b, for real a and b. Take a = 1, b = −1. Then a² = b² = 1, so the assumption holds; but a ≠ b, so the conclusion fails. One example, claim dead. Note x = ½ would not work as a counterexample to “if x² > x then x > 1”: it makes the assumption false, so it tests nothing. The value that works there is x = −1.

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Common questions

Paper 2 logic FAQ

What is the difference between necessary and sufficient?

A is sufficient for B means A implies B: whenever A holds, B holds. B is necessary for A means the same arrow read from the other end: A cannot hold without B. Both labels describe one implication, and confusing the direction is the single most common Paper 2 error.

What does 'A only if B' mean?

'A only if B' means 'if A, then B'. In ordinary speech, 'only if' can sound like a stronger version of 'if', but in mathematics it changes the direction of the implication. B is required for A, so A guarantees B.

What is the contrapositive of a statement?

The contrapositive of 'if A then B' is 'if not B then not A'. It always has the same truth value as the original. The converse, 'if B then A', is not equivalent, and Paper 2 trap options rely on exactly that confusion.

What makes a valid counterexample?

A counterexample to 'if A then B' is a permitted case where A holds and B fails, both at once. An example that breaks the assumption disproves nothing. Try small integers, zero, negatives, fractions and boundary cases before anything elaborate.

How do I negate a quantified statement?

Work from the outer structure inward: 'for all' becomes 'there exists', 'there exists' becomes 'no', then negate the core property exactly. The negation of 'every student scored above 70' is 'at least one student scored 70 or below', not 'no student scored above 70'.