Propositions and their Converses – Class 9 Mathematics notes, important questions and MCQs

Chapter overview

Mathematics builds on clear statements (propositions) that are either true or false. Reversing an "if–then" statement gives its converse, which may or may not be true.

Propositions

• A proposition is a statement that is either true or false, not both.
• "If P, then Q": P is the hypothesis, Q is the conclusion.
• A counter-example is one case that shows a statement is false.

Converse and proof

• Converse of "If P then Q" is "If Q then P".
• Example: "If a triangle is equilateral, its angles are equal" — the converse is also true.
• Example: "If a number is divisible by 4, it is even" — true; converse "If even, divisible by 4" is false (6).
• Theorems are proved from axioms and earlier results using logical steps.

Key terms

Axiom

a basic statement accepted without proof

Theorem

a statement proved using logic

Counter-example

an example that disproves a general statement

Converse

the statement with hypothesis and conclusion swapped

Formulas

• Converse of "P ⇒ Q" is "Q ⇒ P"

Important questions with answers

Quick revision

• A proposition is either true or false.
• Converse swaps P and Q.
• Pythagoras theorem and its converse are both true.

Exam tips

• One counter-example is enough to disprove a statement; examples alone never prove it.
• Questions like "Is the converse true?" need either a proof or a counter-example.

Practice MCQs

1. Which is a proposition?

   1. Close the door.
   2. Is it raining?
   3. 7 is a prime number.
   4. What a nice day!
   Show answer

   Answer: (c) 7 is a prime number.. It is a statement that is true or false.

2. The converse of "If P, then Q" is

   1. If not P, then not Q
   2. If Q, then P
   3. P and Q
   4. If P, then not Q
   Show answer

   Answer: (b) If Q, then P. Swap hypothesis and conclusion.

3. A statement accepted without proof is called

   1. a theorem
   2. a lemma
   3. a converse
   4. an axiom
   Show answer

   Answer: (d) an axiom. Axiom.

4. A counter-example to "All prime numbers are odd" is

   1. 3
   2. 9
   3. 2
   4. 11
   Show answer

   Answer: (c) 2. 2 is prime and even.

5. "If a number is divisible by 6, it is divisible by 3." The converse is

   1. true
   2. false
   3. not a statement
   4. always true
   Show answer

   Answer: (b) false. 9 is divisible by 3 but not 6.

6. In "If two angles are vertically opposite, they are equal", the conclusion is

   1. two angles are vertically opposite
   2. if
   3. angles
   4. they are equal
   Show answer

   Answer: (d) they are equal. The part after "then".

7. The converse of the Pythagoras theorem says

   1. in a right triangle, H² = P² + B²
   2. all triangles are right-angled
   3. if the square of one side equals the sum of squares of the other two, the angle opposite it is 90°
   4. a² + b² > c² always
   Show answer

   Answer: (c) if the square of one side equals the sum of squares of the other two, the angle opposite it is 90°. It identifies a right angle from side lengths.

8. Which statement AND its converse are both true?

   1. If a quadrilateral is a square, it is a rectangle
   2. If x = 2, then x² = 4
   3. If a triangle has three equal sides, it has three equal angles
   4. If n is divisible by 10, n is even
   Show answer

   Answer: (c) If a triangle has three equal sides, it has three equal angles. Equilateral ⇔ equiangular.