TMUA Paper 2: The logic of arguments, set 2

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

1. 11 mark

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

   1. A Every square is a rectangle.
   2. B Every rectangle is a square.
   3. C The sum of the interior angles of any triangle is 180 degrees, except for right-angled triangles.
   4. D If a number is divisible by 6, then it is divisible by 12.
2. 21 mark

   Maya is on the debating team. Maya is not on the chess team.

   Let R be the statement 'Maya is on the debating team' and let S be the statement 'Maya is on the chess team.'

   Using the ordinary meanings of 'and', 'or' and 'not', which of the following is true?

   1. A R and S
   2. B not R and S
   3. C R and not S
   4. D not R and not S
3. 31 mark

   Consider this statement about the integer x = 9: 'If x is even, then x^2 is even.'

   Since 9 is not even, what can we say about the truth of this conditional statement, for x = 9?

   1. A The statement is false for x = 9, because its condition (x is even) is false.
   2. B The statement is true for x = 9: a conditional with a false condition counts as true, whatever the conclusion turns out to be.
   3. C The statement's truth cannot be determined for x = 9, because the condition does not hold.
   4. D The statement is true only if we can also confirm separately that 9^2 = 81 is even.
4. 41 mark

   A password counts as 'strong' if it has at least 12 characters.

   Which of the following correctly restates this as an 'if ... then ...' statement?

   1. A If a password is strong, then it has at least 12 characters.
   2. B A password is strong if and only if it has at least 12 characters.
   3. C If a password has fewer than 12 characters, then it is not strong.
   4. D If a password has at least 12 characters, then it is strong.
5. 51 mark

   A student may sit the resit exam only if they missed the original exam through illness certified by a doctor's note.

   Which of the following correctly restates this as an 'if ... then ...' statement?

   1. A If a student does not sit the resit exam, then they did not miss the original exam through illness certified by a doctor's note.
   2. B If a student sits the resit exam, then they missed the original exam through illness certified by a doctor's note.
   3. C If a student missed the original exam through illness certified by a doctor's note, then they may sit the resit exam.
   4. D A student may sit the resit exam if and only if they missed the original exam through illness certified by a doctor's note.
6. 61 mark

   It is a fact that: 'A positive integer n is a perfect square if and only if it has an odd number of positive divisors.'

   The number m = 20 has 6 positive divisors (1, 2, 4, 5, 10 and 20), which is an even number, not an odd one.

   What can we conclude about m = 20?

   1. A m is definitely not a perfect square.
   2. B m might or might not be a perfect square; having an even number of divisors does not settle this.
   3. C m is definitely a perfect square.
   4. D We cannot conclude anything, because the biconditional only applies to numbers with an odd number of divisors.
7. 71 mark

   Consider the statement: 'If a polygon is a regular hexagon, then it has six equal sides.'

   What is the converse of this statement?

   1. A If a polygon is not a regular hexagon, then it does not have six equal sides.
   2. B If a polygon does not have six equal sides, then it is not a regular hexagon.
   3. C A polygon is a regular hexagon if and only if it has six equal sides.
   4. D If a polygon has six equal sides, then it is a regular hexagon.
8. 81 mark

   Consider the statement: 'If an integer n is divisible by 15, then n is divisible by 3.'

   What is the contrapositive of this statement?

   1. A If n is divisible by 3, then n is divisible by 15.
   2. B If n is not divisible by 15, then n is not divisible by 3.
   3. C If n is not divisible by 3, then n is not divisible by 15.
   4. D n is divisible by 15 if and only if n is divisible by 3.
9. 91 mark

   The statement 'If a quadrilateral is a square, then it has four equal sides' 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 quadrilateral does not have four equal sides, then it is not a square.'
   2. B The converse: 'If a quadrilateral has four equal sides, then it is a square.'
   3. C Both the converse and the contrapositive.
   4. D Neither the converse nor the contrapositive.
10. 101 mark

    For a real number x, consider the condition 'x > 5.'

    Which of the following conditions is necessary, but not sufficient, for x > 5 to hold?

    1. A x > 10
    2. B x < 0
    3. C x - 5 > 0
    4. D x > 0
11. 111 mark

    Which of the following statements is true?

    1. A For all integers n, if n is even then n + 1 is even.
    2. B For some integers n, n^2 = n.
    3. C For all real numbers x, if x^2 = 4 then x = 2.
    4. D There exists a real number x such that x^2 = -4.
12. 121 mark

    Consider the statement: 'There exists an integer n such that n is a multiple of 4 and n is odd.'

    Which of the following is the correct negation of this statement?

    1. A There exists an integer n such that n is not a multiple of 4 and n is not odd.
    2. B For all integers n, n is not a multiple of 4 and n is not odd.
    3. C For all integers n, n is not a multiple of 4 or n is not odd.
    4. D There exists an integer n such that n is a multiple of 4 or n is odd.

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 definition or a small example rather than guessing.
   2. Statement A is true: a square has four right angles and four equal sides, which satisfies the definition of a rectangle (four right angles), so every square is a rectangle.
   3. Statement B fails because a rectangle only needs four right angles, not equal sides; a 2 by 3 rectangle is a rectangle but not a square.
   4. Statement C fails because the angle sum of any triangle, right-angled or not, is always 180 degrees; there is no exception.
   5. Statement D fails because 6 is divisible by 6 but 6 is not divisible by 12, so being divisible by 6 does not guarantee being divisible by 12.
   6. Since B, C and D are all false and A is true, the answer is A.
   • Why not B: Confuses a subset relation for an equivalence: mistakes 'every square is a rectangle' for a two-way relationship, when a rectangle need not have equal sides to be a rectangle - a 2 by 3 rectangle is a counterexample.
   • Why not C: Wrongly believes right-angled triangles are an exception to the angle-sum property, when a right-angled triangle's three angles - one of 90 degrees and two others summing to 90 degrees - still add to 180 degrees overall.
   • Why not D: Reverses the direction of implication, treating a smaller divisor's multiple as automatically a multiple of a larger one; 6 itself is divisible by 6 but not by 12, providing a direct counterexample.

2. Question 2Answer: C

   1. Work out the truth value of R and S first: R, 'Maya is on the debating team', is true, and S, 'Maya is on the chess team', is false.
   2. 'R and not S' needs both R to be true and 'not S' to be true; R is true, and since S is false, 'not S' is true, so 'R and not S' is true overall.
   3. Checking the alternatives: 'R and S' needs S to be true, but S is false, so it is false; 'not R and S' needs R to be false, but R is true, so it is false; 'not R and not S' also needs R to be false, so it too is false.
   4. Only 'R and not S' comes out true, so the answer is C.
   • Why not A: Assumes both membership facts hold simultaneously, forgetting that 'and' requires every part to be true; here S is false since Maya is not on the chess team.
   • Why not B: Mistakenly applies 'not' to R, when Maya is on the debating team, so R is true and 'not R' is false; the whole conjunction fails regardless of S.
   • Why not D: Correctly notes that Maya is not on the chess team but incorrectly also applies 'not' to R, when R is actually true, since Maya is on the debating team; with 'not R' false, the whole conjunction fails.

3. Question 3Answer: B

   1. A conditional statement 'if A then B' is false in exactly one situation: when A is true and B is false. In every other combination of truth values, including whenever A is false, the conditional counts as true.
   2. Here x = 9, so the condition 'x is even' is false.
   3. Because the condition is false, the whole conditional statement is automatically true for x = 9, regardless of whether the conclusion 'x^2 is even' happens to be true or false.
   4. As it happens, 9^2 = 81, which is odd, so the conclusion is actually false here; but this does not matter, because a false condition already guarantees the conditional's truth on its own.
   5. So the statement is true for x = 9, which is option B.
   • Why not A: Mistakes a false condition for making the whole conditional false; a conditional 'if A then B' is only false in the one case where A is true and B is false, so a false A can never make it false.
   • Why not C: Confuses the truth of the whole conditional, which is fully determined, with the truth of the conclusion B alone, which really would be undetermined if we only knew A was false and had no conditional linking A and B.
   • Why not D: Wrongly believes the conditional's truth still depends on separately checking the conclusion's own truth value, when a false condition settles the conditional as true regardless of what the conclusion turns out to be.

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 password is strong' and B is 'it has at least 12 characters', so 'a password is strong if it has at least 12 characters' means 'if a password has at least 12 characters, then it is strong'.
   3. Swapping the two parts instead gives the converse, 'if a password is strong, then it has at least 12 characters', which is not what the original sentence says.
   4. Negating both parts without swapping gives the inverse, which is also not guaranteed to be equivalent; and forcing an 'if and only if' over-claims a second direction the original statement never promised.
   5. So the correct restatement is 'If a password has at least 12 characters, then it is strong', option D.
   • Why not A: Reverses the direction of the conditional, treating 'A if B' as though it read 'A, then B' from left to right rather than 'if B, then A'.
   • Why not B: Turns the statement into a biconditional instead of a simple conditional, over-claiming that having fewer than 12 characters would also guarantee a password is not strong, when the original statement commits to only one direction.
   • Why not C: Gives the inverse of the correct restatement, negating both the character-count condition and the strength conclusion without swapping their order; the inverse is not logically guaranteed to match the original statement.

5. Question 5Answer: B

   1. The phrase 'A only if B' translates to 'if A then B': here A is 'a student sits the resit exam' and B is 'they missed the original exam through illness certified by a doctor's note'.
   2. So the statement means 'if a student sits the resit exam, then they missed the original exam through illness certified by a doctor's note'.
   3. Swapping the two parts instead gives the converse, 'if a student missed the original exam through illness certified by a doctor's note, then they may sit the resit exam', which over-claims a guarantee the original statement never makes.
   4. Negating both parts without swapping gives the inverse, which is also not guaranteed to hold; and forcing an 'if and only if' over-claims a second direction on top of the one the statement actually commits to.
   5. So the correct restatement is 'If a student sits the resit exam, then they missed the original exam through illness certified by a doctor's note', option B.
   • Why not A: Gives the inverse of the correct conditional, negating both the resit-sitting condition and the illness condition without swapping their order; the inverse is not guaranteed to share the truth of the original 'A only if B' statement.
   • Why not C: Reverses the direction of the conditional, treating 'A only if B' as though it read 'if B then A' rather than 'if A then B'; having a certified illness does not by itself guarantee permission to resit, since other conditions might also apply.
   • Why not D: Turns the statement into a biconditional, over-claiming that missing the exam through certified illness would also be enough on its own to guarantee permission to resit, when the original statement only commits to the one direction.

6. Question 6Answer: A

   1. 'If and only if' statements hold in both directions at once: 'perfect square implies odd number of divisors', and also, just as importantly, 'not an odd number of divisors implies not a perfect square'.
   2. m = 20 has 6 divisors, which is an even number, so it does not have an odd number of divisors.
   3. Because the biconditional guarantees that having a non-odd (even) number of divisors implies not being a perfect square, m = 20 cannot be a perfect square.
   4. This checks out directly too: 4^2 = 16 and 5^2 = 25, and 20 lies strictly between them, so 20 is indeed not a perfect square.
   5. So the correct conclusion is that m is definitely not a perfect square, option A.
   • Why not B: Treats the biconditional as though it only guaranteed the 'perfect square implies odd number of divisors' direction, and misses that the reverse direction, an even number of divisors implies not a perfect square, is just as guaranteed by an 'if and only if' statement.
   • Why not C: Directly contradicts the given biconditional, treating an even number of divisors as evidence for being a perfect square, rather than against it.
   • Why not D: Wrongly restricts the biconditional's guarantee to only the case actually mentioned, odd divisors, when 'if and only if' covers every case: whenever one side is false, the other side must be false too, not just when one side happens to be true.

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 polygon is a regular hexagon, then it has six equal sides'; swapping the two parts gives 'if a polygon has six equal sides, then it is a regular hexagon'.
   3. This is different from the inverse, which negates both parts without swapping, and the contrapositive, which negates both parts and swaps them.
   4. The correct converse is 'If a polygon has six equal sides, then it is a regular hexagon', which is option D. Whether the converse itself is true is a separate question from what it is.
   • Why not A: Gives the inverse - negating both the hexagon condition and the equal-sides condition without swapping their order - rather than the converse, which requires swapping without negating.
   • Why not B: Gives the contrapositive - negating both parts and swapping their order - rather than the converse, which swaps without negating.
   • Why not C: Turns the statement into a biconditional instead of simply swapping hypothesis and conclusion, over-claiming the two directions must share the same truth value; a polygon can have six equal sides without being a regular hexagon, since its angles could differ, so the converse here is actually false, even though it is still correctly identified as the converse.

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 n is divisible by 15, then n is divisible by 3'; negating both parts gives 'n is not divisible by 15' and 'n is not divisible by 3', and swapping their order gives 'if n is not divisible by 3, then n is not divisible by 15'.
   3. This is different from the converse, which swaps without negating, and the inverse, which negates without swapping.
   4. The correct contrapositive is 'If n is not divisible by 3, then n is not divisible by 15', which is option C.
   • Why not A: Gives the converse of the original statement, swapping the divisibility conditions 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 two parts; n = 3 is divisible by 3 but not by 15, so the two conditions are not equivalent, and forcing a biconditional over-claims what the original statement guarantees.

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 quadrilateral does not have four equal sides, then it is not a square', must also be true.
   4. The converse, 'if a quadrilateral has four equal sides, then it is a square', is actually false: a rhombus has four equal sides but its angles need not be right angles, so it need not be a square, 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 a rhombus has four equal sides but need not have right angles, so it need not be a square.
   • Why not C: Over-generalises, assuming both related statements automatically inherit the truth of the original, when only the contrapositive is logically guaranteed to.
   • 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 'x > 5' if x > 5 forces the condition to hold; it is sufficient if the condition on its own forces x > 5 to hold.
    2. x > 0 is necessary: whenever x > 5, it is automatically true that x > 0.
    3. x > 0 is not sufficient on its own: x = 1 satisfies x > 0 but not x > 5.
    4. By contrast, x > 10 is sufficient but not necessary, x < 0 is neither necessary nor sufficient, and x - 5 > 0 is just x > 5 rearranged, so it is both necessary and sufficient.
    5. So the condition that is necessary but not sufficient is x > 0, option D.
    • Why not A: This condition is sufficient (x > 10 always gives x > 5) but not necessary (x = 6 satisfies x > 5 without satisfying x > 10), so it has the necessary and sufficient roles the wrong way round.
    • Why not B: This condition is neither necessary nor sufficient: x > 5 can hold while x < 0 fails, so x < 0 is not necessary; and x < 0 can hold while x > 5 fails, so it is not sufficient either.
    • Why not C: This condition is just a rearrangement of x > 5 itself, so it is both necessary and sufficient, not merely necessary.

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. n = 0 satisfies n^2 = n, since 0^2 = 0; this single example is enough to make 'for some integers n, n^2 = n' true.
    3. 'For all integers n, if n is even then n + 1 is even' fails at n = 2, since 2 + 1 = 3 is odd, not even, so it is false.
    4. 'For all real numbers x, if x^2 = 4 then x = 2' fails at x = -2, since (-2)^2 = 4 but -2 is not equal to 2, so it is false.
    5. 'There exists a real number x such that x^2 = -4' is false, because squaring any real number, whether positive, negative or zero, always gives a result of zero or more.
    6. The only true statement is 'For some integers n, n^2 = n', option B.
    • Why not A: Overlooks that an even number plus 1 is always odd, not even; n = 2 gives n + 1 = 3, which is odd, so this 'for all' claim is false as soon as a single even n is tried.
    • Why not C: Overlooks that x = -2 also satisfies x^2 = 4, so the equation has a second solution that is not equal to 2, making the 'for all' claim false.
    • Why not D: Confuses 'there exists' with the very claim being disproved: squaring any real number always gives a result of zero or more, so no real number has a negative square.

12. Question 12Answer: C

    1. To negate a 'there exists' statement, change it to 'for all' and negate what follows; to negate 'P and Q', replace it with 'not P or not Q'.
    2. The original statement is 'there exists an integer n such that n is a multiple of 4 and n is odd'; switching the quantifier gives 'for all integers n...' and negating the inner conjunction gives '...n is not a multiple of 4 or n is not odd'.
    3. Combining these two steps gives 'for all integers n, n is not a multiple of 4 or n is not odd', which correctly denies the original statement.
    4. Any option that keeps the quantifier as 'there exists', or that keeps 'and' instead of switching it to 'or', has failed to apply one of the two negation steps correctly.
    5. So the correct negation is option C.
    • Why not A: Fails to switch the quantifier from 'there exists' to 'for all', which negating an existential statement requires, and also keeps 'and' instead of switching to 'or', which negating a conjunction requires.
    • Why not B: Correctly switches the quantifier to 'for all' but keeps the joining word as 'and' instead of switching to 'or'; negating 'P and Q' gives 'not P or not Q', not 'not P and not Q'.
    • Why not D: Leaves the quantifier as 'there exists' and leaves both inner parts unnegated, only swapping 'and' for 'or'; a correct negation must both switch the quantifier and negate the inner parts.