Proposition P: If it rains, then the road is wet.
Converse Q: If the road is wet, then it has rained.
Step-by-step solution
Idea: To show a statement ‘if X then Y’ is not always true, find one situation where X happens but Y does not. That situation is called a counterexample.
- P says: whenever it rains, the road gets wet. Rain falling on a road makes it wet, so P is true.½ mark
- Q swaps the two parts: whenever the road is wet, it must have rained.½ mark
- But a road can be wet for other reasons. For instance, a tanker may have spilled water on the road.½ mark
- In that case the road is wet but it has not rained. This is a counterexample: a situation that shows why Q is not necessarily true. So P is true, but Q may not be true.½ mark
Answer to write in the exam
P: If it rains, then the road is wet. — True
Q: If the road is wet, then it has rained.
Counterexample: the road is wet because a tanker spilled water on it; it has not rained.
∴ P is true, but its converse Q is not necessarily true.
Common mistakes that cost marks
- Giving a ‘counterexample’ in which the road is dry. A counterexample to Q must make the ‘if’ part happen (the road is wet) and the ‘then’ part fail (no rain).
- Mixing up the converse with ‘If it does not rain, then the road is not wet’. That is a different statement.
- Thinking that because P is true, Q must be true too.
How this can come in the exam
Which situation is a counterexample to ‘If a plant is watered daily, then it flowers’?
- A plant watered daily that flowers
- A plant not watered that does not flower
- A plant watered daily that never flowers
- A plant not watered that flowers
Show answer
(C) A plant watered daily that never flowers
A counterexample must make the ‘if’ part true (watered daily) and the ‘then’ part false (no flowers).
Write the converse of ‘If a person is in Paris, the capital of France, then the person is in France.’ Is the proposition true? Is the converse true? Justify.
Show answer
The proposition is true, since Paris, the capital of France, is in France (½ mark). Converse: ‘If a person is in France, then the person is in Paris, the capital of France’ (½ mark). It is false: a person in Lyon is in France but not in Paris (1 mark).Try one yourself
Write the converse of ‘If a shape is a circle, then it has no corners’. Is the converse true? If not, give a counterexample.
Show answer
Converse: ‘If a shape has no corners, then it is a circle.’ False: an oval has no corners but is not a circle.
More questions like this
- Proposition P: If a number is a multiple of 6, then it is a multiple of 3.
Converse Q: If a number is a multiple of 3, then it is a multiple of 6. - In this example, n is any positive integer.
Proposition P: If n is a perfect square, then it has an odd number of factors.
Converse Q: If n has an odd number of factors, then it is a perfect square.
You may recall that we came across these statements in the previous grade. Which of them are true? - Consider the following argument.
Each factor of a number has a ‘partner’ factor such that their product yields the given number, e.g., 5 is a factor of 35, and 5 × 7 = 35. Here, 7 is the partner factor of 5, and vice-versa. Let us focus on factor-partner pairs of numbers, e.g., (1, 12), (2, 6), (3, 4) are the pairs for the number 12.
If a number has an odd number of factors, there must be a factor-partner pair in which the same number repeats (e.g., the partner factor of 5 in 25). If not, the given number will have an even number of factors since each factor can be paired with its factor pair.
Thus, if a number has an odd number of factors, then it is a perfect square.
What does this argument prove? Statement P or Q? - This only proves Q. It doesn’t prove that every square number has an odd number of factors (Proposition P).
Is P true? - (i) n has an odd number of factors
which implies
(ii) the existence of a factor-partner pair in which the same number repeats, say (f, f)
which implies
(iii) n is a square number: n = f × f
We start with (iii). Clearly, (iii) implies (ii). Does (ii) imply (i)?