10 June 2026

Updated 17 July 2026

TMUA.co.uk team

TMUA Paper 2 Logic: Necessary vs Sufficient, Contrapositive and Counterexamples

Paper 2 adds elementary logic, proof and error-spotting to the full Section 1 mathematics syllabus. This guide focuses on four recurring reasoning errors: mixing up necessary and sufficient, treating a converse as equivalent to the original statement, mishandling quantifier negation, and offering counterexamples that do not satisfy the hypothesis.

Table of contents

Intro

Paper 2 is 20 questions in 75 minutes on mathematical reasoning. It can use every topic from Section 1, then adds elementary logic, mathematical proof and identifying errors in proofs. The extra language is often unfamiliar because A-level courses do not teach it systematically. This post covers the official additional scope, then drills four parts of it through original worked examples.

Want this as a structured reference? The full logic language is laid out on one page, with translation tables, reveal-to-check questions and the free 23-page Logic Decoder PDF, at tmua.co.uk/paper-2-logic.

What Paper 2 actually tests

The official Section 2 content has three strands:

  • The Logic of Arguments (Arg1-Arg4): true and false statements; and, inclusive or, and not; "if", "only if" and "if and only if"; converse and contrapositive; necessary and sufficient conditions; "for all", "for some" and "there exists"; and negating statements built from that language.
  • Mathematical Proof (Prf1-Prf5): direct deductive proof, proof by cases, contradiction and disproof by counterexample; deducing implications; making and justifying conjectures from small cases; arranging statements into a valid proof; and following sophisticated chains of reasoning.
  • Identifying Errors in Proofs (Err1-Err2): locating an error in a purported proof and recognising common invalid deductions.

Candidates are not expected to recognise formal symbolic logic notation or complete formal truth tables. This guide uses P, Q and arrows only as private scratch-work shorthand. For the conditional "if P then Q":

  • Converse: Q ⇒ P. Not equivalent to the original.
  • Contrapositive: not-Q ⇒ not-P. Equivalent to the original.
  • P is sufficient for Q: P ⇒ Q.
  • P is necessary for Q: Q ⇒ P.

Those four translations cover one important part of Arg1 and Arg2, not the whole Paper 2 scope. The examples below also use quantifier negation and counterexamples. They should sit alongside practice in proof by cases, contradiction, conjectures, proof ordering and error identification.

Necessary vs sufficient: the arrow direction

Worked example 1. Let x be a real number. Consider:

  • P: x³ > 27
  • Q: x² > 9

Is P necessary, sufficient, both, or neither for Q?

Translate P first: x³ > 27 is exactly x > 3, because cubing preserves order over the reals. Now check both arrows.

P ⇒ Q: if x > 3 then x² > 9. True. So P is sufficient for Q.

Q ⇒ P: if x² > 9 then x > 3? No: x = -4 gives x² = 16 > 9 but x³ = -64. So P is not necessary for Q.

Answer: sufficient but not necessary. Notice where the work happened: rewriting x³ > 27 as x > 3 (cubing is order-preserving; squaring is not) and then remembering that x² > 9 means x > 3 or x < -3. The exam buries the logic test inside a small algebra test, and the negative branch of the square is the most reused trap in the entire paper.

The translation habit that saves you: "P is necessary for Q" means Q cannot happen without P, so the arrow runs Q ⇒ P. If you have to think about it for more than ten seconds, you have not drilled it enough yet.

Contrapositive vs converse

Worked example 2. A function f has a stationary point at x = a. Consider the true statement:

S: if f′(a) = 0 and f″(a) < 0, then f has a local maximum at a.

Which of the following follow from S?

  1. If f does not have a local maximum at a, then f′(a) ≠ 0 or f″(a) ≥ 0.
  2. If f has a local maximum at a, then f′(a) = 0 and f″(a) < 0.

Statement 1 is the contrapositive of S: not-Q ⇒ not-P, with the "and" in P correctly negated to an "or". Contrapositives are equivalent to the original, so statement 1 follows. Statement 2 is the converse, and it is actually false as a piece of maths: f(x) = -x⁴ has a local maximum at 0 with f″(0) = 0. The second-derivative test is sufficient for a maximum, not necessary.

Two things to take from this. First, the contrapositive of "A and B ⇒ Q" is "not-Q ⇒ not-A or not-B"; negation turns and into or (De Morgan), and trap options routinely keep the and. Second, the converse of a familiar true theorem often sounds true because you have used the theorem a hundred times. Paper 2 exploits that familiarity deliberately.

Counterexamples: both halves or nothing

Worked example 3. Consider the claim:

For every positive integer n, n² + n + 41 is prime.

Which value of n disproves it?

Try the structure first rather than arithmetic: n² + n + 41 = n(n + 1) + 41. If n = 40, this is 40 × 41 + 41 = 41 × 41 = 41², visibly composite. So n = 40 is a counterexample, and you found it by factorisation, not by testing 40 numbers. (The claim really is true for every n from 1 to 39, which is precisely why "I checked a few cases" is not a proof.)

The general discipline: a counterexample to "if P then Q" must satisfy P and fail Q, both, explicitly checked. Paper 2 answer options include candidates that fail P, and they look attractive because they fail Q dramatically. They disprove nothing. A related true-claim trap: to show "f increasing on the reals implies f′(x) > 0 everywhere" is false, f(x) = x³ works, since it is strictly increasing but f′(0) = 0. Strictness lives at single points, and single points are where the TMUA likes to live.

Negating quantified statements

Worked example 4. Consider:

T: for every real number x, there exists a real number y such that y² = x.

What is the negation of T, and is T true?

Negation flips each quantifier and negates the core: there exists a real x such that for every real y, y² ≠ x. And T is false: x = -1 is a witness, since no real y squares to a negative.

The mechanical rule: "for all" becomes "there exists", "there exists" becomes "for all", working outside-in, and only the innermost statement gets negated. The trap options negate the inner statement while leaving the quantifiers alone ("for every x there exists y with y² ≠ x"), which is a different and usually irrelevant statement. If you negate quantifiers by feel rather than by rule, the feel will fail you at question 17 with six minutes left.

The traps, collected

  • Arrow reversal: "necessary" translated as P ⇒ Q instead of Q ⇒ P.
  • Converse smuggling: a converse of a familiar theorem presented as equivalent.
  • And/or under negation: negating "A and B" to "not-A and not-B".
  • Counterexamples that fail the hypothesis: dramatic, irrelevant.
  • The negative branch: x² > 9 read as x > 3.
  • Boundary cases: increasing versus strictly increasing, ≥ versus >, stationary versus turning. Equality cases are where wrong options are grown.

Several of these reappear, alongside their Paper 1 cousins, in 10 TMUA Mistakes That Cost Marks Under Time Pressure; logic is where they concentrate because the wrong options are engineered from them.

How to drill this

Reading this post once will not survive contact with question 14 under time pressure. The content is compact, but the skill needs repeated practice. World 1 of our Logic course is free and starts from first principles; the later worlds require TMUA Pro. The TMUA Pro question bank provides further practice filtered by type, with explanations for the wrong options. Slot the drilling into weeks 9-12 of the plan in How to Prepare for the TMUA in 16 Weeks. If implications still feel slippery after a fortnight of practice, use a focused 1-1 session with a Cambridge student tutor to diagnose the exact translation error.

Closing

Paper 2 rewards precise reasoning across mathematics, logic and proof. Four useful habits are to translate conditions before judging them, separate contrapositives from converses, check both halves of every counterexample, and negate quantifiers by rule. They do not replace the rest of the official scope: direct proof, cases, contradiction, conjectures, proof ordering and error identification also need deliberate practice.

Frequently asked questions

What does TMUA Paper 2 test? Paper 2 can use all Section 1 mathematics and adds logic, mathematical proof and identifying errors in proofs. Its additional scope includes conditionals, converse and contrapositive, necessary and sufficient conditions, quantifiers and negation, direct proof, proof by cases, contradiction, counterexamples, implications, conjectures and valid chains of reasoning. It is 20 questions in 75 minutes, no calculator.

What is the difference between necessary and sufficient? P is sufficient for Q means P implies Q: whenever P holds, Q holds. P is necessary for Q means Q implies P: Q cannot hold without P. The arrow directions are opposite, and confusing them is the single most common Paper 2 error.

What is the contrapositive of a statement? The contrapositive of "if P then Q" is "if not Q then not P". It is logically equivalent to the original statement. The converse, "if Q then P", is not equivalent, and Paper 2 trap options rely on exactly that confusion.

What makes a valid counterexample? A counterexample to "if P then Q" must satisfy P and fail Q, both at once. An example that fails P disproves nothing, however dramatic it looks. Checking both halves explicitly is the discipline the exam rewards.

Is Paper 2 harder than Paper 1? Not intrinsically, but it feels harder to most students because the logic content is not taught at A-level. Students who train it systematically often end up finding Paper 2 the more predictable of the two papers.

Sources: UAT-UK TMUA content specification (Paper 2: mathematical reasoning). All worked examples are original. Last updated 2026-07-17.