It can also happen that both the proposition and converse are false! Can you give an example?
Step-by-step solution
Idea: Choose two properties where neither one always leads to the other. Then the proposition and its converse each have a counterexample.
- Pick two properties where neither always gives the other: ‘is a multiple of 4’ and ‘is a multiple of 6’.½ mark
- Proposition: If a number is a multiple of 4, then it is a multiple of 6. Counterexample: 8 = 4 × 2 is a multiple of 4 but not of 6. So it is false.½ mark
- Converse: If a number is a multiple of 6, then it is a multiple of 4. Counterexample: 6 = 6 × 1 is a multiple of 6 but not of 4. So it is false too.½ mark
- Both are false. A geometry pair works the same way: ‘If a quadrilateral is a rhombus, then it is a rectangle’ (false: a rhombus with angles 60° and 120°) and its converse ‘If a quadrilateral is a rectangle, then it is a rhombus’ (false: a 5 cm × 3 cm rectangle).½ mark
Answer to write in the exam
Proposition: If a number is a multiple of 4, then it is a multiple of 6.
Counterexample: 8 = 4 × 2, but 8 is not a multiple of 6. False.
Converse: If a number is a multiple of 6, then it is a multiple of 4.
Counterexample: 6 = 6 × 1, but 6 is not a multiple of 4. False.
∴ Both the proposition and its converse are false.
Common mistakes that cost marks
- Choosing a pair where one direction is actually true, such as ‘multiple of 4 ⇒ even’ (true). Test both directions.
- Giving one counterexample for both statements. Each false statement needs its own counterexample.
How this can come in the exam
For which proposition are BOTH the proposition and its converse false?
- If n is a multiple of 10, then n is a multiple of 5.
- If n is a multiple of 2, then n is a multiple of 3.
- If n is even, then n + 1 is odd.
- If n is a multiple of 12, then n is a multiple of 3.
Show answer
(B) If n is a multiple of 2, then n is a multiple of 3.
2 is a multiple of 2 but not of 3, and 3 is a multiple of 3 but not of 2. In (A) and (D) the proposition is true; in (C) both directions are true.
Assertion (A): ‘If a triangle is right-angled, then it is isosceles’ and its converse are both false.
Reason (R): A triangle with sides 3, 4, 5 is right-angled but not isosceles, and an equilateral triangle is isosceles but not right-angled.
- Both Assertion (A) and Reason (R) are true and R is the correct explanation of A.
- Both Assertion (A) and Reason (R) are true but R is not the correct explanation of A.
- Assertion (A) is true but Reason (R) is false.
- Assertion (A) is false but Reason (R) is true.
Show answer
(A) Both Assertion (A) and Reason (R) are true and R is the correct explanation of A.
R gives a counterexample to each statement, which is exactly why both are false.
Try one yourself
Show that ‘If a number is prime, then it is odd’ and its converse are both false.
Show answer
2 is prime but even, so the proposition is false. Converse ‘If a number is odd, then it is prime’: 9 is odd but not prime, so it is false too.
More questions like this
- Here is a statement of the Baudhāyana–Pythagoras theorem.
Let a, b, c be the sidelengths of a triangle. If the triangle is right-angled, then a2 + b2 = c2.
What is its converse? Is this converse true? - 1. Identify real-life examples in which a proposition is true but not the converse. Give nice counterexamples!
2. Give more examples from geometry as well as number theory in which a proposition is true but not its converse. Give appropriate counterexamples. - If two lines are parallel, then the corresponding angles formed by a transversal are equal.
- If a quadrilateral is a square, then all its angles are equal.
- Given any ∆ABC, let us bisect the angles at B and C. The bisectors meet at the incentre I of the triangle.
Now extend the bisectors beyond I till they meet the opposite sides at E and F respectively, as shown.
Proposition: If AB = AC, then IE = IF.