TMUA Paper 2: The logic of arguments, set 3

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

1. 11 mark

   Let n = 187. Consider the two statements P: 'n is a prime number' and Q: 'n is a multiple of 11.' Using the ordinary (inclusive) meaning of 'or', exactly one of the following four compound statements about n is true. Which one?

   1. A P and Q
   2. B (not P) and Q
   3. C P or (not Q)
   4. D not (P or Q)
2. 21 mark

   It is claimed that the statement 'if x^2 = 49 then x = 7' is true for every real number x. Which value of x below is a genuine counterexample that shows this claim is false?

   1. A x = 7
   2. B x = 0
   3. C x = 49
   4. D x = -7
3. 31 mark

   Consider the statement: 'A positive integer n is not prime if n is a multiple of 4.' Which of the following is a correct translation of this statement into 'if ... then ...' form?

   1. A If n is a multiple of 4, then n is not prime.
   2. B If n is not prime, then n is a multiple of 4.
   3. C If n is not a multiple of 4, then n is prime.
   4. D n is prime if and only if n is not a multiple of 4.
4. 41 mark

   Take as given the true statement: 'A positive integer n can be written as a sum of two square numbers only if n is not of the form 4k + 3, for some non-negative integer k.' Since 71 = 4 x 17 + 3, what can we conclude about 71?

   1. A 71 can definitely be written as a sum of two square numbers.
   2. B 71 might or might not be a sum of two square numbers; the statement does not tell us.
   3. C 71 cannot be written as a sum of two square numbers.
   4. D The statement does not apply to 71, since 71 is itself a prime number.
5. 51 mark

   Take as given the true statement: 'A parallelogram is a rectangle if and only if its diagonals are equal in length.' A parallelogram R has diagonals that are not equal in length. What can we conclude about R?

   1. A R is definitely a rectangle.
   2. B R is definitely not a rectangle.
   3. C R might or might not be a rectangle; the statement does not tell us.
   4. D The statement does not apply, since R might not have any right angles.
6. 61 mark

   For a real number x, which of the following conditions is necessary, but not sufficient, for x^2 < 9?

   1. A 0 < x < 3
   2. B -3 < x < 3
   3. C x^2 < 4
   4. D x < 3
7. 71 mark

   Which of the following statements is true?

   1. A For all real numbers x, x^2 >= x.
   2. B There exists a real number x such that x^2 = -1.
   3. C For some integers n, n^3 < n^2.
   4. D For all positive integers n, n! > 2^n.
8. 81 mark

   Consider the statement: 'For all real numbers x, if x > 2 then x^2 > 4 and x^3 > 8.' Which of the following is the correct negation of this statement?

   1. A There exists a real number x such that x > 2 and (x^2 <= 4 or x^3 <= 8).
   2. B There exists a real number x such that x > 2 and x^2 <= 4 and x^3 <= 8.
   3. C There exists a real number x such that x <= 2 and (x^2 <= 4 or x^3 <= 8).
   4. D For all real numbers x, if x > 2 then x^2 <= 4 or x^3 <= 8.
9. 91 mark

   Take as given the true statement: 'If a positive integer n is a multiple of 9, then the sum of n's digits is a multiple of 9.' Which option best describes the converse of this statement (if the sum of n's digits is a multiple of 9, then n is a multiple of 9) and whether it is true?

   1. A The converse is automatically true, because a statement and its converse must share the same truth value.
   2. B The converse is automatically false, because a converse is never guaranteed by the original statement.
   3. C The converse is undefined, since 'sum of digits' is not something the original statement discusses in reverse.
   4. D The converse cannot be determined from the original statement alone, but checking examples confirms it is also true here.
10. 101 mark

    Take as given the true statement: 'For all integers n, if n^2 is even then n is even.' Using this statement and its contrapositive, what can we conclude about n^2 if n is odd?

    1. A n^2 must be even.
    2. B n^2 must be odd.
    3. C n^2 could be either even or odd; nothing follows.
    4. D This tells us nothing about odd n, since the statement only discusses even n^2.
11. 111 mark

    Consider the statement: 'There exists a real number x such that x^2 - 6x + 10 <= 0.' Which option gives the correct negation of this statement, together with a correct judgement of whether the negation is true?

    1. A For all real x, x^2 - 6x + 10 > 0; the negation is true, since completing the square gives x^2 - 6x + 10 = (x - 3)^2 + 1, which is always at least 1.
    2. B For all real x, x^2 - 6x + 10 >= 0; the negation is true, by the same reasoning.
    3. C There exists a real x such that x^2 - 6x + 10 > 0; the negation is true.
    4. D For all real x, x^2 - 6x + 10 > 0; the negation is false, since the original statement is true for x = 3.
12. 121 mark

    It is a necessary condition for three positive lengths a, b and c to form a triangle that all three of a + b > c, b + c > a and a + c > b hold. Given a = 3, b = 4 and c = 9, what can we conclude?

    1. A Since two of the three inequalities do hold, these lengths can form a triangle.
    2. B Since a + b = 7 is close to c = 9, these lengths approximately form a triangle.
    3. C Since a necessary condition fails (3 + 4 = 7, which is not greater than 9), these lengths cannot form a triangle.
    4. D Nothing can be concluded, since a necessary condition failing does not rule out the possibility.

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: B

   1. Test 187 for primality properly: since sqrt(187) is just under 14, every prime up to 13 must be checked. 187 is odd, its digit sum 1 + 8 + 7 = 16 is not a multiple of 3, it does not end in 0 or 5, and 187 / 7 is not a whole number, but 187 / 11 = 17 exactly, so 187 = 11 x 17 is composite: P is false.
   2. Since 187 = 11 x 17, it is indeed a multiple of 11, so Q is true.
   3. Evaluate each option with P false and Q true: 'P and Q' is false; 'P or (not Q)' is false or false, which is false, since Q true makes 'not Q' false; 'not (P or Q)' is not (false or true) = not true = false.
   4. '(not P) and Q' is true and true, which is true, since P is false so 'not P' is true, and Q is true.
   5. Only '(not P) and Q' comes out true, so the answer is B.
   • Why not A: Stops a primality check too early, testing only small primes such as 2, 3, 5 and 7 and concluding 187 must be prime, when in fact 187 = 11 x 17; a full check needs every prime up to sqrt(187), which is just under 14, and 11 is one of them.
   • Why not C: Makes an arithmetic slip on the multiplication 11 x 17, for instance reading it as 186 rather than 187, and so wrongly concludes Q is false; this misses that 11 x 17 = 187 exactly, making Q true and 'not Q' false.
   • Why not D: Misapplies the rule for negating 'or': treats 'not (P or Q)' as though it kept the connective unchanged once P and Q have been evaluated, rather than recognising that P or Q is itself true here (since Q is true), so its negation must be false.

2. Question 2Answer: D

   1. A conditional 'if A then B' is false in exactly one situation: when A is true but B is false. Any other combination leaves the conditional true (or, when A is false, vacuously true).
   2. At x = -7, the hypothesis x^2 = 49 is true, since (-7)^2 = 49.
   3. At the same time, the conclusion x = 7 is false, since -7 is not equal to 7.
   4. This is exactly the hypothesis-true, conclusion-false combination that makes a conditional false, so x = -7 is a genuine counterexample to the claim that the statement holds for all real x.
   5. So the answer is D.
   • Why not A: Substituting x = 7 makes both the hypothesis and the conclusion true, so this case confirms the conditional rather than refuting it; a genuine counterexample needs the hypothesis to hold while the conclusion fails, not a case where both already hold.
   • Why not B: Fails to check the hypothesis first: at x = 0, x^2 = 0, which does not equal 49, so the hypothesis is false here and a conditional with a false hypothesis is automatically (vacuously) true, regardless of the conclusion. This case cannot refute the claim.
   • Why not C: Misreads the hypothesis x^2 = 49 as though it meant x = 49 directly, forgetting the squaring; in fact 49^2 = 2401, not 49, so the hypothesis is false at x = 49, making this case irrelevant to refuting the claim, not a counterexample.

3. Question 3Answer: A

   1. The phrase 'A if B' translates directly to 'if B then A': the part after 'if' becomes the hypothesis, and the other part becomes the conclusion.
   2. Here A is 'n is not prime' and B is 'n is a multiple of 4', so 'n is not prime if n is a multiple of 4' becomes 'if n is a multiple of 4, then n is not prime'.
   3. Swapping the two parts instead would give the converse; negating both parts without swapping would give the inverse; and adding a reverse direction would turn it into a biconditional the sentence never claims.
   4. The correct translation, without any of these changes, is 'If n is a multiple of 4, then n is not prime', which is option A.
   • Why not B: Reverses the direction of the implication, translating 'A if B' as though it read 'if A then B' rather than 'if B then A'; this gives the converse of the correct translation, not the translation itself.
   • Why not C: Negates both parts of the correct conditional without swapping them, giving its inverse ('if not B then not A') rather than the original 'if B then A'; an inverse is not logically equivalent to the statement it is drawn from.
   • Why not D: Over-claims a biconditional ('if and only if') where the original statement only asserts a one-directional implication; nothing in the given sentence guarantees the reverse direction as well.

4. Question 4Answer: C

   1. 'A only if B' translates to 'if A then B': here A is 'n can be written as a sum of two squares' and B is 'n is not of the form 4k + 3'.
   2. Since 71 = 4 x 17 + 3, n = 71 is of the form 4k + 3, so 'B' (n is not of that form) is false for n = 71, meaning 'not B' holds.
   3. The contrapositive of 'if A then B' is 'if not B then not A', and a statement and its contrapositive always share the same truth value, so since the original is given true, 'if not B then not A' is guaranteed true as well.
   4. With 'not B' confirmed true for 71, the contrapositive forces 'not A' to be true: 71 cannot be written as a sum of two square numbers.
   5. So the answer is C.
   • Why not A: Misreads 'A only if B' as though it meant 'A if B', reversing the direction of the implication; this wrongly treats 71 being of the form 4k + 3 as supporting rather than ruling out 71 being a sum of two squares.
   • Why not B: Correctly notices that 'only if' gives a one-directional implication, but wrongly assumes this means nothing can be concluded when the consequent fails; in fact the contrapositive of a one-directional statement gives a firm conclusion whenever the consequent is false.
   • Why not D: Introduces a true but irrelevant fact (71 is prime) that plays no part in the given statement's logical structure, distracting from applying the stated implication directly.

5. Question 5Answer: B

   1. An 'if and only if' statement holds in both directions at once: 'rectangle implies equal diagonals' and, just as importantly, 'equal diagonals implies rectangle'.
   2. The second direction has a contrapositive that is equally guaranteed: 'not equal diagonals implies not a rectangle'.
   3. R's diagonals are given as not equal in length, which is exactly the hypothesis of that contrapositive.
   4. So the contrapositive forces the conclusion that R is not a rectangle; this is a definite conclusion, not merely a possibility, and no further information about right angles is needed.
   5. The answer is B.
   • Why not A: Confuses which side of the biconditional is triggered: assumes that because the two properties are linked by 'if and only if', the given fact about the diagonals must force R to be a rectangle, without checking whether the fact given is the presence or the absence of the linked property.
   • Why not C: Treats the biconditional as though it only guaranteed a conclusion in one direction, like an ordinary 'if ... then ...' statement, missing that 'if and only if' also guarantees the reverse: unequal diagonals force the shape to not be a rectangle.
   • Why not D: Introduces a consideration (right angles) that is not one of the two properties the given biconditional actually links, distracting from applying the stated statement directly.

6. Question 6Answer: D

   1. x^2 < 9 holds exactly when -3 < x < 3. A condition is necessary for x^2 < 9 if x^2 < 9 always forces the condition to hold; it is sufficient if the condition alone always forces x^2 < 9.
   2. 'x < 3' is necessary: whenever -3 < x < 3, it is certainly true that x < 3.
   3. 'x < 3' is not sufficient on its own: x = -10 satisfies x < 3, but (-10)^2 = 100, which is not less than 9.
   4. By contrast, '-3 < x < 3' is both necessary and sufficient (it is exactly equivalent), while '0 < x < 3' and 'x^2 < 4' are each sufficient but not necessary, since both exclude values such as x = -1 or x = 2.5 that still satisfy x^2 < 9.
   5. The condition that is necessary but not sufficient is 'x < 3', option D.
   • Why not A: This condition is sufficient (any x with 0 < x < 3 does satisfy x^2 < 9) but not necessary, since a negative value such as x = -1 also satisfies x^2 < 9 without lying in this range; the necessary and sufficient roles have been swapped.
   • Why not B: This condition is actually both necessary and sufficient for x^2 < 9, since it is exactly equivalent to it; picking it as 'necessary but not sufficient' misses that it is also sufficient, over and above what is asked.
   • Why not C: This condition is sufficient (x^2 < 4 does force x^2 < 9) but not necessary, since a value such as x = 2.5 gives x^2 = 6.25, which is less than 9 but not less than 4; again the necessary and sufficient roles have been swapped.

7. Question 7Answer: C

   1. 'For all' statements are false as soon as a single counterexample is found; 'for some' (or 'there exists') statements are true as soon as a single example is found.
   2. For 'for all real x, x^2 >= x': at x = 0.5, x^2 = 0.25 < 0.5, so this is false.
   3. For 'there exists real x with x^2 = -1': a real number squared is never negative, so this is false.
   4. For 'for all positive integers n, n! > 2^n': at n = 1, 1! = 1 and 2^1 = 2, so 1 > 2 is false, refuting the claim immediately.
   5. For 'for some integers n, n^3 < n^2': at n = -1, n^3 = -1 and n^2 = 1, and -1 < 1 is true, so this statement is true.
   6. Only the third statement is true, so the answer is C.
   • Why not A: Tests only integers such as x = 2 or x = -2, where x^2 >= x does hold, and misses a fractional counterexample such as x = 0.5, where x^2 = 0.25, which is less than 0.5; a 'for all real numbers' claim needs every real number checked, not only integers.
   • Why not B: Confuses this with the complex number fact that i^2 = -1, wrongly importing it into a statement restricted to real numbers x; no real number's square can be negative, since a real square is always zero or positive.
   • Why not D: Correctly senses that factorial growth eventually overtakes exponential growth for large n, but fails to test the small cases first: at n = 1, 1! = 1 while 2^1 = 2, so 1! > 2^1 is already false, which alone refutes the 'for all' claim.

8. Question 8Answer: A

   1. Negating a 'for all ... if A then B' statement requires three separate changes: switch 'for all' to 'there exists'; rewrite 'if A then B' as 'A and not B', keeping A unchanged and negating only B; and, if B is itself a conjunction, negate it with De Morgan's law, switching 'and' to 'or'.
   2. The quantifier becomes 'there exists a real number x such that ...'.
   3. The hypothesis x > 2 is kept exactly as it is, since only the conclusion is negated.
   4. The conclusion 'x^2 > 4 and x^3 > 8' negates to 'x^2 <= 4 or x^3 <= 8', switching the connective as De Morgan's law requires.
   5. Putting the three parts together gives 'there exists a real number x such that x > 2 and (x^2 <= 4 or x^3 <= 8)', which is option A.
   • Why not B: Correctly flips the quantifier and correctly keeps the hypothesis x > 2 unchanged, but misapplies the rule for negating an 'and': negating 'x^2 > 4 and x^3 > 8' should switch the connective to 'or', not leave it as 'and'.
   • Why not C: Correctly flips the quantifier and correctly negates the inner 'and' to 'or', but wrongly negates the hypothesis x > 2 as well; negating 'if A then B' should keep A true and negate only B, not negate both sides.
   • Why not D: Correctly negates the inner 'and' to 'or', but fails to flip the quantifier from 'for all' to 'there exists', and leaves the overall conditional structure in place instead of converting it to an 'and' statement, which is what negating 'if A then B' requires.

9. Question 9Answer: D

   1. A statement and its converse do not automatically share a truth value; only a statement and its contrapositive are guaranteed to do that.
   2. So the converse here (digit sum a multiple of 9 implies n a multiple of 9) is not settled just by the truth of the original statement; it needs to be checked on its own.
   3. Testing examples: 18 has digit sum 9, a multiple of 9, and 18 is indeed a multiple of 9; 4,509 has digit sum 4 + 5 + 0 + 9 = 18, a multiple of 9, and 4,509 / 9 = 501, confirming it too.
   4. In this particular case the converse does hold (this is in fact the standard divisibility test for 9, which works in both directions), but the key point is that this had to be checked rather than assumed from the original statement's truth.
   5. The answer is D.
   • Why not A: Applies the rule that only holds for the contrapositive (which always shares the original statement's truth value) to the converse instead, which has no such guarantee and must be checked independently.
   • Why not B: Over-corrects the opposite way: having correctly noticed that a converse is not automatically guaranteed, wrongly concludes it can therefore never be true, when in fact a converse is simply undetermined by the logic alone and can turn out either way.
   • Why not C: Invents a technicality that swapping the hypothesis and conclusion of a conditional is somehow undefined here; forming the converse of any 'if A then B' statement is always well-defined, regardless of what A and B describe.

10. Question 10Answer: B

    1. The contrapositive of 'if n^2 is even then n is even' is 'if n is not even (that is, n is odd) then n^2 is not even (that is, n^2 is odd)'.
    2. A statement and its contrapositive always share the same truth value, and the original statement is given as true, so this contrapositive is also true for every integer n.
    3. n is given to be odd, which is exactly the hypothesis of the contrapositive.
    4. So the contrapositive's conclusion applies directly: n^2 must be odd.
    5. The answer is B.
    • Why not A: Misreads which direction applies, defaulting to the original statement's conclusion ('even') without correctly forming and applying the contrapositive to the case of an odd n.
    • Why not C: Forgets that a statement and its contrapositive always share the same truth value; since the original statement is given as true, its contrapositive is guaranteed true too, so a firm conclusion does follow, not an undetermined one.
    • Why not D: Wrongly restricts the statement's scope to only the literal case named in its hypothesis (even n^2), missing that a universally quantified statement's contrapositive applies directly to the odd case as well.

11. Question 11Answer: A

    1. Negating a 'there exists ... such that (expression) <= 0' statement gives 'for all ... , (expression) > 0': the quantifier flips to 'for all', and the inequality '<=' flips to the strict '>'.
    2. Completing the square: x^2 - 6x + 10 = x^2 - 6x + 9 + 1 = (x - 3)^2 + 1.
    3. Since (x - 3)^2 is never negative, (x - 3)^2 + 1 is always at least 1, which is always strictly greater than 0, for every real x.
    4. So the negation, 'for all real x, x^2 - 6x + 10 > 0', is true, which also means the original statement was false all along.
    5. The answer is A.
    • Why not B: Negates '<=' (less than or equal to) to '>=' instead of the strict '>' that is required; the negation of 'less than or equal to' is 'strictly greater than', not 'greater than or equal to', which would wrongly re-include the equality case that must be excluded.
    • Why not C: Correctly flips the inequality from '<=' to '>' but fails to flip the quantifier from 'there exists' to 'for all', which negating a 'there exists' statement always requires.
    • Why not D: Forms the negation correctly but evaluates its truth wrongly: substituting x = 3 into the original expression gives 3^2 - 6(3) + 10 = 9 - 18 + 10 = 1, which is not less than or equal to 0, so x = 3 does not in fact satisfy the original statement.

12. Question 12Answer: C

    1. A necessary condition must hold if the conclusion is to be true; if a necessary condition fails, the conclusion is definitely false, with no exceptions.
    2. Check the three inequalities with a = 3, b = 4, c = 9: b + c > a gives 4 + 9 = 13 > 3, true; a + c > b gives 3 + 9 = 12 > 4, true; but a + b > c gives 3 + 4 = 7 > 9, which is false.
    3. Since all three inequalities are stated as necessary together, and one of them (a + b > c) fails, the necessary condition as a whole fails.
    4. A failed necessary condition guarantees the conclusion cannot hold, regardless of how close the failing inequality came to being true, so these lengths cannot form a triangle.
    5. The answer is C.
    • Why not A: Wrongly treats satisfying a majority of several necessary conditions as good enough, not realising that all of them must hold at once for the conclusion to be possible; failing even one of the three is already enough to rule it out completely.
    • Why not B: Imports an approximation mindset from measurement into a strict mathematical inequality; 7 > 9 is simply false, with no partial credit for being numerically close, so this does not weaken the conclusion at all.
    • Why not D: Reverses the roles of necessary and sufficient conditions: it is a sufficient condition's failure that tells us nothing on its own, whereas a necessary condition's failure is exactly what does rule the conclusion out completely.