Most tested Paper 2 (Mathematical Reasoning)

The Logic of Arguments: Conditionals, Converse, Contrapositive and Quantifiers

On TMUA Paper 2 you reason precisely in words: reading "if⋯then" statements, forming converses and contrapositives, telling necessary from sufficient conditions, handling "for all" and "there exists", and negating each correctly. No symbols or truth tables are needed - just quick, careful logic.

Part of the TMUA syllabus — revision for the Test of Mathematics for University Admission (TMUA), the UAT-UK maths test used by Cambridge, Oxford, Imperial, UCL, LSE, Warwick and Durham. No calculator; multiple choice.

Key points

  • A statement is a sentence that is definitely true or definitely false; 'A and B' needs both parts true, inclusive 'A or B' is false only when both parts are false, and 'not' flips the truth value.
  • 'If A then B' claims that whenever A holds, B also holds; it says nothing about the case where A is false, and it does NOT claim that B forces A.
    A ⇒ B
  • The converse of 'if A then B' is 'if B then A' - a genuinely different statement, so a true claim can easily have a false converse.
  • The contrapositive of 'if A then B' is 'if not B then not A', and it ALWAYS has the same truth value as the original, so proving either one proves both.
  • Read the phrasings carefully: 'A if B' means 'if B then A', 'A only if B' means 'if A then B', and 'A if and only if B' means both directions hold at once.
  • In 'if A then B', A is sufficient for B (A guarantees B) and B is necessary for A (A cannot hold without B); 'necessary and sufficient' is exactly 'if and only if'.
  • 'For all x' is disproved by a single counterexample, while 'there exists x' / 'for some x' needs just one example to be true; 'for some' means at least one, and does not rule out all.
  • Negation rules: not(A and B) = (not A) or (not B); not(if A then B) = A and (not B); and not(for all x, P) = there exists x with not P.

Formulae

contrapositive: (if A then B) has the same truth value as (if not B then not A)

Prove or test a hard 'if A then B' by switching to the easier 'if not B then not A' - they stand or fall together.

converse: (if A then B) does NOT force (if B then A)

Reject any answer that assumes a statement and its converse must both be true.

not(if A then B) = A and (not B)

Negate a conditional, or build a counterexample: find a case where A is true but B is false.

not(for all x, P(x)) = there exists x with not P(x)

Negate a universal claim; a single counterexample makes a 'for all' statement false.

not(there exists x, P(x)) = for all x, not P(x)

Negate an existence claim; you must rule out every x, not just one.

not(A and B) = (not A) or (not B) ; not(A or B) = (not A) and (not B)

De Morgan's laws: push a 'not' through an 'and'/'or', flipping the connective as you go.

Definitions

Converse
The converse of 'if A then B' is 'if B then A', formed by swapping the hypothesis and conclusion. It is a separate statement and need not share the original's truth value.
Contrapositive
The contrapositive of 'if A then B' is 'if not B then not A', formed by swapping AND negating both parts. It always has the same truth value as the original statement.
Sufficient condition
'A is sufficient for B' means A guarantees B: if A is true then B is true. A on its own is enough to force B, though B might also arise other ways.
A ⇒ B
Necessary condition
'A is necessary for B' means B cannot hold without A: if B is true then A is true. A is required for B, but A alone may not be enough to give B.
B ⇒ A

Worked examples

1

Consider the statement: 'For every integer n, if n is divisible by 6 then n is divisible by 3.' Write its converse and its contrapositive, and decide which of the three statements are true.

  1. 1

    Identify the parts.

    The hypothesis A is 'n is divisible by 6' and the conclusion B is 'n is divisible by 3'.

  2. 2

    The original is true, since any multiple of 6 is also a multiple of 3.

  3. 3

    Write:

    6k = 3(2k)
  4. 4

    This shows a multiple of 6 is a multiple of 3, so 'if A then B' is true.

  5. 5

    Form the contrapositive by swapping and negating:

    'if n is NOT divisible by 3 then n is NOT divisible by 6'.

  6. 6

    The contrapositive always matches the original's truth value, so it is also true.

  7. 7

    Form the converse by swapping only:

    'if n is divisible by 3 then n is divisible by 6'.

  8. 8

    Test the converse with a small case.

    Take n = 3
  9. 9
    Here n = 3 is divisible by 3 but not by 6, so the converse is false

Answer: Converse: 'if n is divisible by 3 then n is divisible by 6' (false). Contrapositive: 'if n is not divisible by 3 then n is not divisible by 6' (true). The original and its contrapositive are true; the converse is false, with n = 3 as a counterexample.

2

Write the negation of the statement: 'For every real number x, if x > 3 then x2 > 9.' Then decide whether the original statement or its negation is true.

  1. 1

    The statement has the shape 'for all x, if A then B', with A being x > 3 and B being x2 > 9.

  2. 2

    Negate the quantifier:

    'for all' becomes 'there exists'.

  3. 3

    Negate the inner conditional using not(if A then B) = A and (not B):

    this gives 'x > 3 and x2 ≤ 9'.

  4. 4

    So the negation reads:

    'there exists a real number x with x > 3 and x2 ≤ 9'.

  5. 5

    Now test the original.

    Since x > 3, x is positive, so multiplying x > 3 by x keeps the inequality.

  6. 6

    Since x > 3:

    x × x > 3 × x.

  7. 7

    Also:

    3 × x > 9.

  8. 8

    So:

    x2 > 9.

  9. 9

    Every x > 3 really does give x2 > 9, so no x satisfies the negation.

Answer: Negation: 'there exists a real number x with x > 3 and x2 ≤ 9.' The original statement is true, so this negation is false - no such x exists.

Common mistakes

  • ×Confusing a statement with its converse: 'if A then B' being true does NOT make 'if B then A' true (the classic 'affirming the consequent' slip).
  • ×Misreading 'A only if B' as 'if B then A'. It actually means 'if A then B', so B is the necessary condition for A.
  • ×Negating 'if A then B' as 'if A then not B'. The correct negation is 'A is true AND B is false' - a single counterexample, not another conditional.
  • ×Negating 'all X are Y' as 'all X are not Y'. The correct negation is 'some X is not Y', which needs just one counterexample.
  • ×Mixing up necessary and sufficient: a sufficient condition guarantees the result, a necessary one is merely required, and only 'if and only if' gives both.

No-calculator tips

  • ✓To knock out a 'for all' statement, hunt for one counterexample; the usual suspects are 0, 1, negatives, and fractions - test them fast in your head.
  • ✓To disprove 'if A then B', find one case with A true but B false; that single case is also exactly the negation, so it does double duty.
  • ✓When a question gives 'if A then B', jot the contrapositive too - they share a truth value, so test whichever is easier and both are settled.
  • ✓Translate every worded phrasing into 'if ⋯ then ⋯' before reasoning: 'A only if B' → 'if A then B', 'A if B' → 'if B then A', and iff → both directions.
  • ✓When you negate, always swap 'for all' with 'there exists', and keep the inclusive reading of 'or' and the 'at least one' reading of 'for some'.

Test yourself

Original practice questions, no calculator. Work each out before revealing the answer.

Q1.Consider the true statement about a positive integer n: "If n is divisible by 6, then n is divisible by 3." Which of the following is the CONTRAPOSITIVE of this statement?

  • A. If n is divisible by 3, then n is divisible by 6.
  • B. If n is not divisible by 6, then n is not divisible by 3.
  • C. If n is not divisible by 3, then n is not divisible by 6.
  • D. If n is not divisible by 3, then n is divisible by 6.
  • E. If n is divisible by 6, then n is not divisible by 3.
Show answer

Answer: C — If n is not divisible by 3, then n is not divisible by 6.

The contrapositive of "P implies Q" is "not Q implies not P", so both parts are negated AND swapped: not(div by 3) implies not(div by 6). Option 0 is the converse (swap only) and option 1 is the inverse (negate only).

Q2.For a real number x, compare the condition "x > 3" with the condition "x > 1". Which statement is correct?

  • A. "x > 3" is necessary but not sufficient for "x > 1".
  • B. "x > 3" is sufficient but not necessary for "x > 1".
  • C. "x > 3" is both necessary and sufficient for "x > 1".
  • D. "x > 3" is neither necessary nor sufficient for "x > 1".
  • E. "x > 1" is sufficient for "x > 3".
Show answer

Answer: B — "x > 3" is sufficient but not necessary for "x > 1".

x > 3 guarantees x > 1 (so it is sufficient), but x > 1 does not force x > 3, e.g. x = 2 (so it is not necessary). Option 0 reverses which condition implies which.

Q3.A bag contains several balls. Consider the statement "All of the balls in the bag are red." Which of the following is the correct NEGATION of this statement?

  • A. No ball in the bag is red.
  • B. All of the balls in the bag are not red.
  • C. At least one ball in the bag is red.
  • D. At least one ball in the bag is not red.
  • E. Exactly one ball in the bag is not red.
Show answer

Answer: D — At least one ball in the bag is not red.

The negation of "all are red" is "it is not the case that all are red", i.e. at least one is not red. Options 0 and 1 over-negate to "none are red", which is a stronger (different) claim, not the negation.

Read this topic in the official UAT-UK TMUA content specification →

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