New NCERT Ganita Manjari · Class 9 Maths · Part II · Chapter 9: Propositions and their Converses
This chapter explains what a proposition is and how most mathematical statements take the form 'If X, then Y'. It shows how to form the converse of a proposition, why a true proposition does not guarantee a true converse, and how a single counterexample can prove a claim false.
From the Class 9 Maths Formula Sheet, made as per the new NCERT book Ganita Manjari · Pages 48–49 of the book.
Key points
- A proposition is a statement that is either true or false, not both. '7 is an even number' is a false proposition.
- Commands and questions, like 'Close the door', are not propositions.
- Most propositions in maths have the form 'If X, then Y', where X is the condition and Y is the result.
- 'If X, then Y', 'X implies Y', 'X ⇒ Y' and 'Y when X' all mean the same.
- The converse of 'If X, then Y' is 'If Y, then X': swap the condition and the result.
- A true proposition does not guarantee a true converse; the two must be checked separately.
- A proposition and its converse can be: one true and one false, both true, or both false.
- A counterexample satisfies the condition but not the result; one counterexample is enough to prove a claim false.
- To prove a proposition true, give reasoning that works for every case; examples alone are not a proof.
- A number is a perfect square exactly when it has an odd number of factors, so the two statements imply each other (X ⇔ Y).
- The Baudhāyana–Pythagoras theorem and its converse are both true.
- Euler showed 2³² + 1 = 641 × 6700417 is composite, a famous counterexample to Fermat's claim.
Formulas
| Converse | Converse of 'If X, then Y' is 'If Y, then X' |
| Both directions true | X ⇔ Y each implies the other |
| Baudhāyana–Pythagoras theorem and converse | right-angled triangle ⇔ a² + b² = c² c is opposite the right angle |
Key terms
- Proposition: A statement that is either true or false, but not both.
- Condition and result: In 'If X, then Y', X is the condition and Y is the result.
- Converse: The statement made by swapping the condition and the result.
- Counterexample: An example that satisfies the condition but not the result, showing the proposition is false.
- Factor-partner pair: A pair (a, b) of factors of n with a × b = n.
Diagrams to learn
- If X then Y (condition implies result) diagram
- Table: proposition vs converse, all four cases
- Factor pairs of 36 and 12
- Triangles for the proof of the Pythagoras converse
- Counterexample bank table
- A true proposition does not guarantee a true converse. 'If it rains, the road is wet' is true, but the converse is not always true.
- One counterexample is enough to prove a claim false, but examples alone do not prove a claim true.
- 'Same area ⇒ congruent' is false: base 4, height 3 and base 6, height 2 both give area 6 sq units.
A real page from the book
Get the full chapter in the book
This page is a summary. The Class 9 Maths Formula Sheet has the complete chapter in colour, with every diagram and worked example.
Frequently asked questions
What is the converse of a proposition?
The converse of 'If X, then Y' is 'If Y, then X'. For example, the converse of 'If it rains, then the road is wet' is 'If the road is wet, then it has rained'.
What is a counterexample in maths?
A counterexample is an example that satisfies the condition but not the result. For 'If n is odd, then n is prime', n = 9 is a counterexample.
Is the converse of the Pythagoras theorem true?
Yes. If a² + b² = c² for the sides of a triangle, then the triangle is right-angled, with the right angle opposite c.
← Part I · Chapter 8: Predicting What Comes Next: Exploring Sequences and ProgressionsPart II · Chapter 10: How Quantities Combine: Understanding Data →
