Revision Libraryv1.6.8
Choose a subject →
Search
WorksheetsBooksFreePracticeHow it worksPricingFreeAboutFAQ
Choose a subject →

Admissions tests / TMUA / Paper 2 / The logic of arguments

Test standard. 12 questions, 12 marks, about 45 minutes.

TMUA Paper 2: The logic of arguments, set 1

True and false, and, inclusive or, not, conditional statements in all their English forms, converse and contrapositive, necessary and sufficient, quantifiers, and negation.

Download the questions (PDF) Download with worked solutions (PDF)

  • Answer all questions. No calculator.
  • Each question has exactly one correct answer.
  1. 11 mark

    Exactly one of the following four statements about whole numbers is true. Which one is it?

    1. A The product of two even numbers is always even.
    2. B 51 is a prime number.
    3. C The sum of two odd numbers is always odd.
    4. D Every multiple of 4 is also a multiple of 8.
  2. 21 mark

    Let n = 12. Consider these two statements about n:

    P: n is a multiple of 5.

    Q: n is a multiple of 3.

    Using the ordinary (inclusive) meaning of 'or', which of the following is true?

    1. A P and Q
    2. B P or Q
    3. C not Q
    4. D not P and not Q
  3. 31 mark

    Consider the statement: 'If Ben read the memo, then he knows about the meeting.' Ben did not read the memo. Which of the following can we correctly conclude, based only on the statement above?

    1. A Ben definitely knows about the meeting.
    2. B Ben definitely does not know about the meeting.
    3. C Ben might or might not know about the meeting; the statement does not tell us.
    4. D The statement above must be false, because Ben is a counterexample.
  4. 41 mark

    Consider the statement: 'The bus is cancelled if it snows.' Which of the following means the same thing?

    1. A If the bus is cancelled, then it snows.
    2. B If it does not snow, then the bus is not cancelled.
    3. C The bus is cancelled only if it snows.
    4. D If it snows, then the bus is cancelled.
  5. 51 mark

    Consider the statement: 'You may enter the exam hall only if you have your candidate number.' Which of the following is a correct conclusion, given that a student has entered the exam hall?

    1. A The student might or might not have their candidate number.
    2. B The student definitely has their candidate number.
    3. C The student definitely does not have their candidate number.
    4. D Having a candidate number guarantees entry to the exam hall.
  6. 61 mark

    Consider the true statement: 'A triangle is equilateral if and only if all three of its angles are equal.' A triangle T has all three angles equal to 60 degrees. What can we conclude about T?

    1. A T is definitely equilateral.
    2. B T might or might not be equilateral; having equal angles does not guarantee this.
    3. C T is definitely not equilateral.
    4. D We can conclude T is equilateral only if we also check its side lengths directly.
  7. 71 mark

    Consider the statement: 'If a number is a multiple of 12, then it is a multiple of 4.' What is the converse of this statement?

    1. A If a number is not a multiple of 12, then it is not a multiple of 4.
    2. B If a number is not a multiple of 4, then it is not a multiple of 12.
    3. C A number is a multiple of 12 if and only if it is a multiple of 4.
    4. D If a number is a multiple of 4, then it is a multiple of 12.
  8. 81 mark

    Consider the statement: 'If a shape is a square, then it has four right angles.' What is the contrapositive of this statement?

    1. A If a shape has four right angles, then it is a square.
    2. B If a shape is not a square, then it does not have four right angles.
    3. C If a shape does not have four right angles, then it is not a square.
    4. D A shape is a square if and only if it has four right angles.
  9. 91 mark

    The statement 'If a number is a multiple of 6, then it is a multiple of 2' is true. Based only on the general relationship between a statement, its converse and its contrapositive, which of the following must also be true?

    1. A The contrapositive: 'If a number is not a multiple of 2, then it is not a multiple of 6.'
    2. B The converse: 'If a number is a multiple of 2, then it is a multiple of 6.'
    3. C Both the converse and the contrapositive.
    4. D Neither the converse nor the contrapositive.
  10. 101 mark

    For a positive integer n, consider the condition 'n is a multiple of 10.' Which of the following is necessary, but not sufficient, for n to be a multiple of 10?

    1. A n is a multiple of 20.
    2. B n is a multiple of 2 and a multiple of 5.
    3. C n ends in the digit 0.
    4. D n is a multiple of 5.
  11. 111 mark

    Which of the following statements is true?

    1. A For all prime numbers p, p is odd.
    2. B For some prime numbers p, p is even.
    3. C For all integers n, n^2 is greater than n.
    4. D There exists an integer n such that n^2 is negative.
  12. 121 mark

    Consider the statement: 'For all triangles, if the triangle is equilateral, then all its angles are 60 degrees.' Which of the following is the correct negation of this statement?

    1. A For all triangles, if the triangle is not equilateral, then not all its angles are 60 degrees.
    2. B There exists a triangle that is not equilateral and has all angles equal to 60 degrees.
    3. C There exists a triangle that is equilateral and does not have all angles equal to 60 degrees.
    4. D For all triangles, the triangle is equilateral and does not have all angles equal to 60 degrees.

Worked solutions

Every question below carries the reasoning, not just the answer. The official material for this test publishes a correct option letter and nothing else.

  1. Question 1Answer: A

    1. Check each statement against a small example rather than guessing: any even number can be written as 2 times a whole number, so multiplying two even numbers gives 2m x 2n = 4mn, which is even; this makes statement A true.
    2. Statement B fails because 51 = 3 x 17, so 51 is not prime.
    3. Statement C fails because two odd numbers sum to an even number, for example 3 + 5 = 8, not an odd number.
    4. Statement D fails because 4 itself is a multiple of 4 but not a multiple of 8, so being a multiple of 4 does not guarantee being a multiple of 8.
    5. Since B, C and D are all false and A is true, the answer is A.
    • Why not B: Thinks 51 is prime, missing that 51 = 3 x 17; checking divisibility only by small primes such as 2, 5 and 7 while overlooking 3 is a common gap (the digit sum 5 + 1 = 6 is divisible by 3, which is a quick way to spot the factor).
    • Why not C: Assumes two numbers of the same type (here, odd) always combine to give another number of that same type, without testing an example such as 3 + 5 = 8, which is even, not odd.
    • Why not D: Confuses the direction of the multiple relationship: every multiple of 8 is a multiple of 4, but the reverse does not hold, since 4 itself is a multiple of 4 but not of 8.

  2. Question 2Answer: B

    1. Work out the truth value of each individual statement first: P says 12 is a multiple of 5, which is false, since 12 / 5 is not a whole number.
    2. Q says 12 is a multiple of 3, which is true, since 12 / 3 = 4.
    3. 'P or Q', using the ordinary, inclusive meaning of 'or', is true whenever at least one of P and Q is true, so with P false and Q true, 'P or Q' is true.
    4. Checking the alternatives: 'P and Q' needs both to be true but P is false, so it is false; 'not Q' is false since Q is true; 'not P and not Q' needs Q to be false, but Q is true, so it is also false.
    5. Only 'P or Q' comes out true, so the answer is B.
    • Why not A: P is false, since 12 is not a multiple of 5, so treating 'P and Q' as true here confuses 'and' with 'or': 'and' needs both parts to be true, not just one of them.
    • Why not C: Q is true, since 12 is a multiple of 3 (12 / 3 = 4), so 'not Q' is false; this option comes from wrongly believing 12 is not a multiple of 3.
    • Why not D: This needs both 'not P' and 'not Q' to be true, but Q is true, so 'not Q' is false; this again comes from wrongly treating 12 as not a multiple of 3.

  3. Question 3Answer: C

    1. The statement 'if Ben read the memo, then he knows about the meeting' only makes a claim about what happens when the 'if' part (reading the memo) is true.
    2. Ben not reading the memo means the 'if' part of the statement simply does not apply to him, so the statement places no requirement on whether Ben knows about the meeting.
    3. Ben could still know about the meeting some other way, for example if someone told him, or he might not know at all; the given statement does not settle this either way.
    4. So the only conclusion the statement supports is that Ben's knowledge of the meeting is undetermined, which is option C.
    • Why not A: Wrongly treats a conditional with a false 'if' part as though it were vacuously true AND that this made the 'then' part itself true, confusing the (correct) fact that 'if Ben read the memo, then he knows' is not contradicted by Ben's not reading it with the (incorrect) claim that his knowing is therefore guaranteed.
    • Why not B: Commits the fallacy of denying the antecedent: assumes that because reading the memo leads to knowing, not reading it must lead to not knowing, when the statement never rules out learning about the meeting some other way.
    • Why not D: Treats a case where the 'if' part is false as though it broke the statement; a statement of the form 'if A then B' only makes a claim when A is true, so a case where A is false cannot be a counterexample to it.

  4. Question 4Answer: D

    1. The phrase 'A if B' translates directly to 'if B then A': the condition after 'if' becomes the hypothesis, and the other part becomes the conclusion.
    2. Here A is 'the bus is cancelled' and B is 'it snows', so 'the bus is cancelled if it snows' means 'if it snows, then the bus is cancelled'.
    3. Swapping the two parts instead gives the converse, 'if the bus is cancelled, then it snows', which is not what the original sentence says; negating both parts without swapping gives the inverse, also not equivalent; reading 'only if' in place of 'if' reverses the direction again.
    4. So the correct rewording is 'If it snows, then the bus is cancelled', which is option D.
    • Why not A: Reverses the direction of the conditional, treating 'A if B' as though it read left to right as 'A, then B' rather than 'if B, then A'.
    • Why not B: Takes the inverse of the correct conditional, negating both the snow condition and the cancellation without swapping them, which is not logically equivalent to the original statement.
    • Why not C: Confuses 'A if B' with 'A only if B'; 'only if' reverses which side becomes the condition, giving the wrong direction of implication.

  5. Question 5Answer: B

    1. The phrase 'A only if B' translates to 'if A then B': here A is 'you may enter the exam hall' and B is 'you have your candidate number', so the statement means 'if you enter the exam hall, then you have your candidate number'.
    2. The student has entered the exam hall, so A is true, and the statement guarantees that B, having the candidate number, must also be true.
    3. This is a direct, certain conclusion rather than merely a possibility, and it is not the reverse condition either.
    4. So the correct conclusion is that the student definitely has their candidate number, which is option B.
    • Why not A: Doubts that 'only if' fully guarantees the condition, when in fact 'you may enter only if you have your candidate number' means entering the hall can only happen when the candidate number condition already holds, so entering fully guarantees it.
    • Why not C: Reverses the direction of the implication, treating 'only if' as ruling out the very condition it actually requires, rather than requiring it.
    • Why not D: Wrongly reads the statement as a biconditional, additionally claiming that having a candidate number is enough on its own to guarantee entry, when the statement never claims this reverse direction.

  6. Question 6Answer: A

    1. 'If and only if' statements hold in both directions at once: 'equilateral implies equal angles' and, just as importantly, 'equal angles implies equilateral'.
    2. Triangle T has all three angles equal to 60 degrees, which is exactly the condition on the right-hand side of the biconditional.
    3. Because the 'if' direction of the biconditional (equal angles implies equilateral) is guaranteed to hold, T must be equilateral; no further checking of side lengths is needed, since the biconditional already covers that.
    4. So the correct conclusion is that T is definitely equilateral, option A.
    • Why not B: Treats the biconditional as only its 'only if' direction, equilateral implies equal angles, and misses that the 'if' direction, equal angles implies equilateral, is also guaranteed by an 'if and only if' statement.
    • Why not C: Confuses the biconditional with its negation, incorrectly treating equal angles as evidence against the triangle being equilateral.
    • Why not D: Doubts that 'if and only if' is a complete logical equivalence, wrongly insisting on an extra, independent check of the side lengths even though the given biconditional already guarantees it.

  7. Question 7Answer: D

    1. The converse of a statement is formed by swapping the hypothesis and the conclusion, without negating either of them.
    2. The original statement is 'if a number is a multiple of 12, then it is a multiple of 4'; swapping the two parts gives 'if a number is a multiple of 4, then it is a multiple of 12'.
    3. This is different from the inverse, which negates both parts without swapping, and the contrapositive, which negates both parts and swaps them, so care is needed not to confuse the three.
    4. The correct converse is 'If a number is a multiple of 4, then it is a multiple of 12', which is option D.
    • Why not A: Gives the inverse of the original statement, negating both parts without swapping them, rather than its converse.
    • Why not B: Gives the contrapositive of the original statement, negating both parts and swapping them, rather than its converse, mixing up the two.
    • Why not C: Turns the statement into a biconditional instead of simply swapping the hypothesis and conclusion, over-claiming that the original statement and its converse must have the same truth value.

  8. Question 8Answer: C

    1. The contrapositive of a statement is formed by negating both the hypothesis and the conclusion, and then swapping their order.
    2. The original statement is 'if a shape is a square, then it has four right angles'; negating both parts gives 'the shape is not a square' and 'it does not have four right angles', and swapping their order gives 'if a shape does not have four right angles, then it is not a square'.
    3. This is different from the converse, which swaps without negating, and the inverse, which negates without swapping.
    4. The correct contrapositive is 'If a shape does not have four right angles, then it is not a square', which is option C.
    • Why not A: Gives the converse of the original statement, swapping hypothesis and conclusion without negating them, rather than its contrapositive.
    • Why not B: Gives the inverse of the original statement, negating both parts without swapping them, rather than its contrapositive, mixing up the two.
    • Why not D: Turns the statement into a biconditional instead of negating and swapping the hypothesis and conclusion, over-claiming that having four right angles is also enough to guarantee being a square.

  9. Question 9Answer: A

    1. A statement and its contrapositive always share the same truth value: whenever one is true, so is the other, and whenever one is false, so is the other.
    2. A statement and its converse do not have this guarantee; the converse can be true or false independently of the original statement.
    3. Here the original statement is true, so its contrapositive, 'if a number is not a multiple of 2, then it is not a multiple of 6', must also be true; this can be checked directly, since any number that is not a multiple of 2 (an odd number) cannot be a multiple of 6 either, as every multiple of 6 is even.
    4. The converse, 'if a number is a multiple of 2, then it is a multiple of 6', is actually false: 2 itself is a multiple of 2 but not of 6, confirming that the converse need not follow.
    5. So only the contrapositive is guaranteed to be true, which is option A.
    • Why not B: Wrongly assumes the converse must share the truth value of the original statement; only the contrapositive is guaranteed to do this. Here the converse is actually false, since 2 is a multiple of 2 but not of 6.
    • Why not C: Over-generalises, assuming both related statements automatically inherit the truth of the original, when only the contrapositive is logically guaranteed to do so.
    • Why not D: Wrongly assumes that negating and swapping a true statement must break its truth, when the contrapositive of a true statement is always true.

  10. Question 10Answer: D

    1. A condition is necessary for 'n is a multiple of 10' if being a multiple of 10 forces the condition to hold; it is sufficient if the condition on its own forces n to be a multiple of 10.
    2. Being a multiple of 5 is necessary: any multiple of 10 is automatically a multiple of 5, since 10 = 2 x 5.
    3. Being a multiple of 5 is not sufficient on its own: n = 5 is a multiple of 5 but not a multiple of 10, since 5 / 10 is not a whole number.
    4. By contrast, being a multiple of 20 is sufficient but not necessary, and being a multiple of both 2 and 5, or ending in the digit 0, are both necessary and sufficient, since either is exactly equivalent to being a multiple of 10.
    5. So the condition that is necessary but not sufficient is 'n is a multiple of 5', option D.
    • Why not A: This condition is actually sufficient (any multiple of 20 is also a multiple of 10) but not necessary (10 itself is a multiple of 10 but not of 20), so it has the necessary and sufficient roles the wrong way round.
    • Why not B: This condition is actually both necessary and sufficient for being a multiple of 10, since 2 and 5 share no common factor, so being a multiple of both is exactly the same as being a multiple of 10, and it wrongly appears to fall short of sufficiency.
    • Why not C: This condition is also both necessary and sufficient for being a multiple of 10 in the ordinary decimal system, so it wrongly appears to fall short of sufficiency.

  11. Question 11Answer: B

    1. 'For all' statements are false if even a single counterexample can be found, while 'for some', or 'there exists', statements are true as soon as a single example can be found.
    2. The number 2 is both prime and even, so it is a counterexample to 'for all prime numbers p, p is odd', making that statement false, and at the same time it is an example confirming that 'for some prime numbers p, p is even' is true.
    3. 'For all integers n, n^2 is greater than n' fails at n = 0 and at n = 1, so it is false.
    4. 'There exists an integer n such that n^2 is negative' is false, because squaring any real number, whether positive, negative or zero, always gives a result of zero or more.
    5. The only true statement is 'For some prime numbers p, p is even', option B.
    • Why not A: Overlooks that 2 is a prime number and is even, so a single counterexample already makes this 'for all' claim false.
    • Why not C: Only checks larger positive integers, where n^2 is greater than n does hold, and misses boundary values such as n = 0, where 0^2 = 0, not greater than 0, or n = 1, where 1^2 = 1, not greater than 1.
    • Why not D: Confuses 'there exists' with the very claim being disproved: squaring any real integer always gives a result of zero or more, so no integer has a negative square.

  12. Question 12Answer: C

    1. To negate a 'for all' statement, change it to 'there exists' and negate what follows; to negate an 'if A then B' statement, replace it with 'A and not B', keeping the hypothesis true but denying the conclusion.
    2. The original statement is 'for all triangles, if the triangle is equilateral, then all its angles are 60 degrees'; negating the quantifier gives 'there exists a triangle such that...' and negating the inner conditional gives '...the triangle is equilateral and not all its angles are 60 degrees'.
    3. Combining these two steps gives 'there exists a triangle that is equilateral and does not have all angles equal to 60 degrees', which correctly denies the original statement.
    4. Any option that keeps the quantifier as 'for all', or that negates the antecedent instead of the consequent, has failed to apply one of the two negation steps correctly.
    5. So the correct negation is option C.
    • Why not A: Forms the inverse of the inner conditional (negating both the equilateral condition and the angle condition, without turning it into an 'and' statement) instead of correctly negating 'if equilateral then all angles 60' to 'equilateral and not all angles 60', and also leaves the quantifier as 'for all' instead of switching it to 'there exists', as negating a 'for all' statement requires.
    • Why not B: Negates the wrong part of the conditional: it should keep the antecedent (equilateral) and negate only the consequent (all angles 60 degrees), rather than negating the antecedent and leaving the consequent unchanged.
    • Why not D: Correctly negates the inner conditional to 'equilateral and not all angles 60 degrees' but fails to switch the quantifier from 'for all' to 'there exists', leaving a statement that could never be true for every triangle.

More on this strand

More free TMUA practice

Every strand of the published TMUA specification, with worked solutions throughout.

Choose a subject →