Question
Is the converse of a true statement always true? Justify your answer with an example.
Solution — Step by Step
For a conditional statement “If P, then Q” (written ), the converse is “If Q, then P” (written ).
The converse simply swaps the hypothesis and conclusion.
Original: If P, then Q Converse: If Q, then P
No — the converse of a true statement is NOT always true.
The truth value of a conditional and its converse are independent. A statement can be true while its converse is false.
True statement: “If a number is divisible by 6, then it is divisible by 2.”
- Is this true? Yes — any multiple of 6 (6, 12, 18, 24…) is even.
Its converse: “If a number is divisible by 2, then it is divisible by 6.”
- Is this true? No. Counterexample: 4 is divisible by 2, but — not divisible by 6.
The original statement is true but its converse is false.
True statement: “If a quadrilateral is a square, then it is a rectangle.”
- True: every square is a rectangle (has four right angles).
Converse: “If a quadrilateral is a rectangle, then it is a square.”
- False: a 4 cm × 6 cm rectangle has four right angles but is NOT a square.
Again, the original is true but the converse is false.
Conclusion: The converse of a true statement is NOT always true. It may be true or false independently.
Why This Works
A conditional says: “being P is sufficient for Q.” But the converse says: “being P is also necessary for Q.” These are different claims.
In the divisibility example: being divisible by 6 is sufficient for divisibility by 2 (true). But it’s not necessary — you can be divisible by 2 without being divisible by 6. The converse confuses sufficiency with necessity.
When BOTH the statement AND its converse are true, we say (if and only if). But this is a special, stronger condition — not the default.
Alternative Method — Counterexample Approach
To disprove a statement, a single counterexample suffices.
Claim: “If a triangle is equilateral, then all angles are 60°.” (TRUE)
Converse claim: “If all angles are 60°, then the triangle is equilateral.”
Is this true? Yes — if all angles are 60°, the triangle must be equilateral (all sides equal). So here the converse happens to also be true.
This illustrates that conversen CAN be true — but are not automatically true.
Common Mistake
Confusing converse with contrapositive. The contrapositive of is (not Q implies not P). The contrapositive is ALWAYS logically equivalent to the original statement — if the original is true, the contrapositive is true. But the converse () is NOT equivalent to the original. Boards frequently test whether students know the difference between converse (not necessarily true) and contrapositive (always equivalent).