TMUA: Converse and contrapositive

Paper 2: if-then statements, the converse and the contrapositive.

Practise converse and contrapositive →Timed TMUA paper

What the specification covers

Key ideas

Common mistakes

Exam tip

Translate every statement into 'if ... then ...' form before comparing.

Worked example

Worked example

What is the contrapositive of the statement: 'If \(n^2\) is odd, then \(n\) is odd'? (\(n\) is an integer.)

  1. If \(n\) is even, then \(n^2\) is odd.
  2. \(n^2\) is odd only if \(n\) is even.
  3. If \(n\) is odd, then \(n^2\) is odd.
  4. If \(n^2\) is even, then \(n\) is even.
  5. If \(n\) is even, then \(n^2\) is even.

Answer: E: If \(n\) is even, then \(n^2\) is even.

The contrapositive of 'if A then B' is 'if not B then not A'. Here A is '\(n^2\) is odd' and B is '\(n\) is odd', so it is 'if \(n\) is even (not odd), then \(n^2\) is even'. Option 2 is the converse; option 3 is the converse's contrapositive (the inverse).

Practice questions

Try each question before opening the solution. No calculator: the TMUA does not allow one. Every answer here was re-checked independently by computer before it was published.

Question 1

Consider the statement: 'A whole number is prime only if it is odd.' Which of the following statements are equivalent to it?
I   If a whole number is prime, then it is odd.
II   If a whole number is odd, then it is prime.
III   If a whole number is even, then it is not prime.

  1. II and III only
  2. none of them
  3. III only
  4. I only
  5. I and II only
  6. II only
  7. I, II and III
  8. I and III only
Show the answer and solution

Answer: H: I and III only

'A only if B' means 'if A then B', which is I. III is the contrapositive of I, so it is equivalent too. II is the converse, which is not equivalent. (The statement itself is false, because 2 is prime; that does not affect which forms are equivalent.)

Question 2

Which of the following statements are true for every real number \(x\)?
I   \(x>1\) if \(x^2>1\).
II   \(x>1\) only if \(x^2>1\).
III   \(x^3>1\) if and only if \(x>1\).

  1. I and III only
  2. II and III only
  3. III only
  4. II only
  5. I and II only
  6. I only
  7. I, II and III
  8. none of them
Show the answer and solution

Answer: B: II and III only

I says 'if \(x^2>1\) then \(x>1\)': false for \(x=-2\). II says 'if \(x>1\) then \(x^2>1\)': true. III: \(x\mapsto x^3\) is strictly increasing, so \(x^3>1\iff x>1\): true.

Question 3

Which of the following statements about positive integers \(n\) have a true converse?
I   If \(n\) is a multiple of 4, then \(n\) is even.
II   If \(n\) is a multiple of 6, then \(n\) is a multiple of 2 and a multiple of 3.
III   If \(n\) is a prime greater than 2, then \(n\) is odd.

  1. II only
  2. I and III only
  3. I and II only
  4. II and III only
  5. I, II and III
  6. none of them
  7. I only
  8. III only
Show the answer and solution

Answer: A: II only

Converse of I: 'even \(\Rightarrow\) multiple of 4', false (\(n=2\)). Converse of II: 'multiple of 2 and 3 \(\Rightarrow\) multiple of 6', true as 2 and 3 are coprime. Converse of III: 'odd \(\Rightarrow\) prime greater than 2', false (\(n=9\)).

Question 4

Consider the statement: 'If a quadrilateral is a square, then it has equal diagonals.' Which of the following is its converse?

  1. If a quadrilateral is not a square, then it does not have equal diagonals.
  2. If a quadrilateral is a square, then it does not have equal diagonals.
  3. If a quadrilateral does not have equal diagonals, then it is not a square.
  4. If a quadrilateral does not have equal diagonals, then it is a square.
  5. If a quadrilateral has equal diagonals, then it is a square.
Show the answer and solution

Answer: E: If a quadrilateral has equal diagonals, then it is a square.

For 'if P then Q' the converse is 'if Q then P' and the contrapositive is 'if not Q then not P' (the contrapositive is equivalent to the statement; the converse is not). Here P is 'a quadrilateral is a square' and Q is 'it has equal diagonals', so the converse is: If a quadrilateral has equal diagonals, then it is a square.

Question 5

Consider the statement: 'If a triangle is equilateral, then it is isosceles.' Which of the following is its converse?

  1. If a triangle is not equilateral, then it is not isosceles.
  2. If a triangle is not isosceles, then it is not equilateral.
  3. If a triangle is isosceles, then it is equilateral.
  4. If a triangle is equilateral, then it is not isosceles.
  5. If a triangle is not isosceles, then it is equilateral.
Show the answer and solution

Answer: C: If a triangle is isosceles, then it is equilateral.

For 'if P then Q' the converse is 'if Q then P' and the contrapositive is 'if not Q then not P' (the contrapositive is equivalent to the statement; the converse is not). Here P is 'a triangle is equilateral' and Q is 'it is isosceles', so the converse is: If a triangle is isosceles, then it is equilateral.

Question 6

Consider the statement: 'If a parallelogram is a rectangle, then it has a right angle.' Which of the following is its converse?

  1. If a parallelogram is not a rectangle, then it has no right angle.
  2. If a parallelogram has a right angle, then it is a rectangle.
  3. If a parallelogram has no right angle, then it is a rectangle.
  4. If a parallelogram is a rectangle, then it has no right angle.
  5. If a parallelogram has no right angle, then it is not a rectangle.
Show the answer and solution

Answer: B: If a parallelogram has a right angle, then it is a rectangle.

For 'if P then Q' the converse is 'if Q then P' and the contrapositive is 'if not Q then not P' (the contrapositive is equivalent to the statement; the converse is not). Here P is 'a parallelogram is a rectangle' and Q is 'it has a right angle', so the converse is: If a parallelogram has a right angle, then it is a rectangle.

Question 7

Consider the statement: 'If an integer is a multiple of 10, then it is even.' Which of the following is its converse?

  1. If an integer is odd, then it is not a multiple of 10.
  2. If an integer is even, then it is a multiple of 10.
  3. If an integer is a multiple of 10, then it is odd.
  4. If an integer is odd, then it is a multiple of 10.
  5. If an integer is not a multiple of 10, then it is odd.
Show the answer and solution

Answer: B: If an integer is even, then it is a multiple of 10.

For 'if P then Q' the converse is 'if Q then P' and the contrapositive is 'if not Q then not P' (the contrapositive is equivalent to the statement; the converse is not). Here P is 'an integer is a multiple of 10' and Q is 'it is even', so the converse is: If an integer is even, then it is a multiple of 10.

Question 8

Consider the statement: 'If a real number is greater than 5, then it has a square greater than 25.' Which of the following is its converse?

  1. If a real number is greater than 5, then it has a square of at most 25.
  2. If a real number has a square of at most 25, then it is greater than 5.
  3. If a real number has a square greater than 25, then it is greater than 5.
  4. If a real number has a square of at most 25, then it is at most 5.
  5. If a real number is at most 5, then it has a square of at most 25.
Show the answer and solution

Answer: C: If a real number has a square greater than 25, then it is greater than 5.

For 'if P then Q' the converse is 'if Q then P' and the contrapositive is 'if not Q then not P' (the contrapositive is equivalent to the statement; the converse is not). Here P is 'a real number is greater than 5' and Q is 'it has a square greater than 25', so the converse is: If a real number has a square greater than 25, then it is greater than 5.

Question 9

Consider the statement: 'If a whole number is a prime greater than 2, then it is odd.' Which of the following is its contrapositive?

  1. If a whole number is even, then it is a prime greater than 2.
  2. If a whole number is odd, then it is a prime greater than 2.
  3. If a whole number is a prime greater than 2, then it is even.
  4. If a whole number is even, then it is not a prime greater than 2.
  5. If a whole number is not a prime greater than 2, then it is even.
Show the answer and solution

Answer: D: If a whole number is even, then it is not a prime greater than 2.

For 'if P then Q' the converse is 'if Q then P' and the contrapositive is 'if not Q then not P' (the contrapositive is equivalent to the statement; the converse is not). Here P is 'a whole number is a prime greater than 2' and Q is 'it is odd', so the contrapositive is: If a whole number is even, then it is not a prime greater than 2.

Question 10

Consider the statement: 'If a triangle is equilateral, then it is isosceles.' Which of the following is its contrapositive?

  1. If a triangle is not equilateral, then it is not isosceles.
  2. If a triangle is isosceles, then it is equilateral.
  3. If a triangle is equilateral, then it is not isosceles.
  4. If a triangle is not isosceles, then it is equilateral.
  5. If a triangle is not isosceles, then it is not equilateral.
Show the answer and solution

Answer: E: If a triangle is not isosceles, then it is not equilateral.

For 'if P then Q' the converse is 'if Q then P' and the contrapositive is 'if not Q then not P' (the contrapositive is equivalent to the statement; the converse is not). Here P is 'a triangle is equilateral' and Q is 'it is isosceles', so the contrapositive is: If a triangle is not isosceles, then it is not equilateral.

Question 11

Consider the statement: 'If a quadrilateral is a square, then it has equal diagonals.' Which of the following is its contrapositive?

  1. If a quadrilateral has equal diagonals, then it is a square.
  2. If a quadrilateral does not have equal diagonals, then it is a square.
  3. If a quadrilateral is a square, then it does not have equal diagonals.
  4. If a quadrilateral does not have equal diagonals, then it is not a square.
  5. If a quadrilateral is not a square, then it does not have equal diagonals.
Show the answer and solution

Answer: D: If a quadrilateral does not have equal diagonals, then it is not a square.

For 'if P then Q' the converse is 'if Q then P' and the contrapositive is 'if not Q then not P' (the contrapositive is equivalent to the statement; the converse is not). Here P is 'a quadrilateral is a square' and Q is 'it has equal diagonals', so the contrapositive is: If a quadrilateral does not have equal diagonals, then it is not a square.

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