Propositions and their Converses
Class 09 MathsA proposition is a statement that is true or false. The converse of the proposition ‘if P then Q’ is the proposition ‘if Q then P’.
Example
Statement 1: If two sides of a triangle are equal, then the angles opposite the equal sides are equal.
Such a statement that is either true or false is called a proposition.
Statement 2: If two angles of a triangle are equal, then the sides opposite the equal angles have equal lengths.
The first proposition is of the form ‘if X then Y’. The second is of the form ‘if Y then X’. We say that the second proposition is the converse of the first proposition.
It can happen that a proposition is true while its converse is false. It can also happen that both the proposition and its converse are true, or both are false.
Counterexamples
Counterexamples are instances that demonstrate the falsity of a given proposition.
Example
- Proposition P: If it rains, then the road is wet.
- Converse Q: If the road is wet, then it has rained.
Proposition P is true but its converse, Q, may not be true. For instance, the road may be wet because a tanker spilled water on the road.
A situation that illustrates why a given proposition is not necessarily true is called a counterexample. It refers to an example that contradicts the stated proposition.