- A proposition is a declarative statement that has a definite truth value, true or false.
- A sentence with an unspecified variable may not yet have a fixed truth value.
- 'If P, then Q' is an implication, written P⇒Q. P is the hypothesis and Q the conclusion.
- The converse of P⇒Q is Q⇒P. The converse need not have the same truth value.
- A universal claim is false if even one example satisfies its hypothesis but not its conclusion.
- The negation of a statement P asserts that P is not true. Negation reverses truth value.
- The contrapositive of P⇒Q is not Q⇒not P; it is logically equivalent to the original.
- The inverse of P⇒Q is not P⇒not Q; it need not be equivalent to the original.
- P is sufficient for Q when P⇒Q; Q is necessary for P. If both directions hold, each is necessary and sufficient.
- P⇔Q means both P⇒Q and Q⇒P; this is a biconditional.
- P⇒Q is false only when P is true and Q false; otherwise it is true in classical logic.
- Start from the hypothesis and use definitions or established facts to derive the conclusion.
Use these notes alongside your current textbook. Check important formulas, worked answers and syllabus coverage with your teacher or official learning material.
Propositions and their Converses — Class 9 Mathematics
Textbook edition: Ganita Manjari 2026–27, Part II, Chapter 9. This belongs to the 2026–27 Ganita Manjari chapter sequence.
Summary: A proposition is a declarative statement that has a definite truth value, true or false.
These are original detailed supplementary notes and practice sets. Exact official exercise numbering, textbook diagrams and activity-by-activity solutions have not been verified against the full prescribed NCERT PDF.
Learning outcomes
- Define terms and state the conditions for formulas and theorems.
- Follow mathematical arguments step by step.
- Apply concepts to new examples and check the results.
- Identify common mistakes and invalid assumptions.
What is a proposition?
A proposition is a declarative statement that has a definite truth value, true or false.
Worked example
'Every square is a rectangle' is true.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Open sentences
A sentence with an unspecified variable may not yet have a fixed truth value.
Worked example
'x>3' depends on the value or domain of x.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Conditional statements
'If P, then Q' is an implication, written P⇒Q. P is the hypothesis and Q the conclusion.
Worked example
If n is divisible by 4, then n is even.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Converse
The converse of P⇒Q is Q⇒P. The converse need not have the same truth value.
Worked example
If n is even, then n is divisible by 4 is false; n=6 disproves it.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Counterexamples
A universal claim is false if even one example satisfies its hypothesis but not its conclusion.
Worked example
'All prime numbers are odd' is false because 2 is prime and even.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Negation
The negation of a statement P asserts that P is not true. Negation reverses truth value.
Worked example
Negation of 'all students passed' is 'at least one student did not pass'.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Contrapositive
The contrapositive of P⇒Q is not Q⇒not P; it is logically equivalent to the original.
Worked example
If divisible by 4 then even; if odd then not divisible by 4.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Inverse
The inverse of P⇒Q is not P⇒not Q; it need not be equivalent to the original.
Worked example
Not divisible by 4 does not imply odd: 6 is a counterexample.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Necessary and sufficient
P is sufficient for Q when P⇒Q; Q is necessary for P. If both directions hold, each is necessary and sufficient.
Worked example
Being divisible by 4 is sufficient for being even, not necessary.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
If and only if
P⇔Q means both P⇒Q and Q⇒P; this is a biconditional.
Worked example
An integer is even iff it is divisible by 2.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Truth of an implication
P⇒Q is false only when P is true and Q false; otherwise it is true in classical logic.
Worked example
'If 2 is even, then 2 is odd' is false.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Proof by direct reasoning
Start from the hypothesis and use definitions or established facts to derive the conclusion.
Worked example
If n=2k, then n²=4k²=2(2k²), so n² is even.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Proof by contrapositive
Prove not Q⇒not P instead of P⇒Q when it is easier.
Worked example
If n² is odd, n is odd, equivalently if n is even, n² is even.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Proof by contradiction
Assume the claim false and derive an impossibility using established facts.
Worked example
A contradiction might be that an integer is simultaneously even and odd.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Geometric converses
Many geometry theorems have distinct converses requiring separate proof.
Worked example
If a quadrilateral is a rectangle it has equal diagonals; equal diagonals alone do not make every quadrilateral a rectangle.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Quantifiers
'All' and 'some' have different logical meanings. Negation of 'all' is 'at least one not'; negation of 'some' is 'none'.
Worked example
Not all triangles are right triangles means at least one triangle is not right.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Common mistakes
Do not assume a true statement has a true converse; test with counterexamples and distinguish converse from contrapositive.
Worked example
A square is a rhombus; a rhombus need not be a square.
Reasoning check
Identify the assumptions and explain why the example satisfies the rule. Where possible, confirm the answer by a second method.
Concepts and formulas at a glance
| Topic | Explanation | Illustration |
|---|---|---|
| What is a proposition? | A proposition is a declarative statement that has a definite truth value, true or false. | 'Every square is a rectangle' is true. |
| Open sentences | A sentence with an unspecified variable may not yet have a fixed truth value. | 'x>3' depends on the value or domain of x. |
| Conditional statements | 'If P, then Q' is an implication, written P⇒Q. P is the hypothesis and Q the conclusion. | If n is divisible by 4, then n is even. |
| Converse | The converse of P⇒Q is Q⇒P. The converse need not have the same truth value. | If n is even, then n is divisible by 4 is false; n=6 disproves it. |
| Counterexamples | A universal claim is false if even one example satisfies its hypothesis but not its conclusion. | 'All prime numbers are odd' is false because 2 is prime and even. |
| Negation | The negation of a statement P asserts that P is not true. Negation reverses truth value. | Negation of 'all students passed' is 'at least one student did not pass'. |
| Contrapositive | The contrapositive of P⇒Q is not Q⇒not P; it is logically equivalent to the original. | If divisible by 4 then even; if odd then not divisible by 4. |
| Inverse | The inverse of P⇒Q is not P⇒not Q; it need not be equivalent to the original. | Not divisible by 4 does not imply odd: 6 is a counterexample. |
| Necessary and sufficient | P is sufficient for Q when P⇒Q; Q is necessary for P. If both directions hold, each is necessary and sufficient. | Being divisible by 4 is sufficient for being even, not necessary. |
| If and only if | P⇔Q means both P⇒Q and Q⇒P; this is a biconditional. | An integer is even iff it is divisible by 2. |
| Truth of an implication | P⇒Q is false only when P is true and Q false; otherwise it is true in classical logic. | 'If 2 is even, then 2 is odd' is false. |
| Proof by direct reasoning | Start from the hypothesis and use definitions or established facts to derive the conclusion. | If n=2k, then n²=4k²=2(2k²), so n² is even. |
| Proof by contrapositive | Prove not Q⇒not P instead of P⇒Q when it is easier. | If n² is odd, n is odd, equivalently if n is even, n² is even. |
| Proof by contradiction | Assume the claim false and derive an impossibility using established facts. | A contradiction might be that an integer is simultaneously even and odd. |
| Geometric converses | Many geometry theorems have distinct converses requiring separate proof. | If a quadrilateral is a rectangle it has equal diagonals; equal diagonals alone do not make every quadrilateral a rectangle. |
| Quantifiers | 'All' and 'some' have different logical meanings. Negation of 'all' is 'at least one not'; negation of 'some' is 'none'. | Not all triangles are right triangles means at least one triangle is not right. |
| Common mistakes | Do not assume a true statement has a true converse; test with counterexamples and distinguish converse from contrapositive. | A square is a rhombus; a rhombus need not be a square. |
Practice set A — Definitions and foundations
- Define what is a proposition? and create a correct example that satisfies all necessary conditions.
- Define open sentences and create a correct example that satisfies all necessary conditions.
- Define conditional statements and create a correct example that satisfies all necessary conditions.
- Define converse and create a correct example that satisfies all necessary conditions.
- Define counterexamples and create a correct example that satisfies all necessary conditions.
- Define negation and create a correct example that satisfies all necessary conditions.
- Define contrapositive and create a correct example that satisfies all necessary conditions.
- Define inverse and create a correct example that satisfies all necessary conditions.
- Define necessary and sufficient and create a correct example that satisfies all necessary conditions.
- Define if and only if and create a correct example that satisfies all necessary conditions.
- Define truth of an implication and create a correct example that satisfies all necessary conditions.
- Define proof by direct reasoning and create a correct example that satisfies all necessary conditions.
Practice set B — Applications
- Verify this worked example: 'Every square is a rectangle' is true. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: 'x>3' depends on the value or domain of x. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: If n is divisible by 4, then n is even. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: If n is even, then n is divisible by 4 is false; n=6 disproves it. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: 'All prime numbers are odd' is false because 2 is prime and even. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: Negation of 'all students passed' is 'at least one student did not pass'. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: If divisible by 4 then even; if odd then not divisible by 4. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: Not divisible by 4 does not imply odd: 6 is a counterexample. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: Being divisible by 4 is sufficient for being even, not necessary. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: An integer is even iff it is divisible by 2. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: 'If 2 is even, then 2 is odd' is false. Change the numerical values or premises and solve the new problem with complete reasoning.
- Verify this worked example: If n=2k, then n²=4k²=2(2k²), so n² is even. Change the numerical values or premises and solve the new problem with complete reasoning.
Practice set C — Higher-order reasoning
- Explain a central theorem or formula and its assumptions.
- Find a counterexample to an incorrect generalisation.
- Compare two concepts and explain their differences.
- Write a real-world application with relevant units or conditions.
- Identify and correct a plausible error in a solution.
- Verify a result using an alternative approach.
- Create an original multi-step problem and solve it.
- Explain the concept to a beginner using an accurate diagram or table.
Frequently asked questions with answers
Explain what is a proposition? with an example.
A proposition is a declarative statement that has a definite truth value, true or false. 'Every square is a rectangle' is true.
Explain open sentences with an example.
A sentence with an unspecified variable may not yet have a fixed truth value. 'x>3' depends on the value or domain of x.
Explain conditional statements with an example.
'If P, then Q' is an implication, written P⇒Q. P is the hypothesis and Q the conclusion. If n is divisible by 4, then n is even.
Explain converse with an example.
The converse of P⇒Q is Q⇒P. The converse need not have the same truth value. If n is even, then n is divisible by 4 is false; n=6 disproves it.
Explain counterexamples with an example.
A universal claim is false if even one example satisfies its hypothesis but not its conclusion. 'All prime numbers are odd' is false because 2 is prime and even.
Explain negation with an example.
The negation of a statement P asserts that P is not true. Negation reverses truth value. Negation of 'all students passed' is 'at least one student did not pass'.
Explain contrapositive with an example.
The contrapositive of P⇒Q is not Q⇒not P; it is logically equivalent to the original. If divisible by 4 then even; if odd then not divisible by 4.
Explain inverse with an example.
The inverse of P⇒Q is not P⇒not Q; it need not be equivalent to the original. Not divisible by 4 does not imply odd: 6 is a counterexample.
Explain necessary and sufficient with an example.
P is sufficient for Q when P⇒Q; Q is necessary for P. If both directions hold, each is necessary and sufficient. Being divisible by 4 is sufficient for being even, not necessary.
Explain if and only if with an example.
P⇔Q means both P⇒Q and Q⇒P; this is a biconditional. An integer is even iff it is divisible by 2.
Explain truth of an implication with an example.
P⇒Q is false only when P is true and Q false; otherwise it is true in classical logic. 'If 2 is even, then 2 is odd' is false.
Explain proof by direct reasoning with an example.
Start from the hypothesis and use definitions or established facts to derive the conclusion. If n=2k, then n²=4k²=2(2k²), so n² is even.
Explain proof by contrapositive with an example.
Prove not Q⇒not P instead of P⇒Q when it is easier. If n² is odd, n is odd, equivalently if n is even, n² is even.
Explain proof by contradiction with an example.
Assume the claim false and derive an impossibility using established facts. A contradiction might be that an integer is simultaneously even and odd.
Explain geometric converses with an example.
Many geometry theorems have distinct converses requiring separate proof. If a quadrilateral is a rectangle it has equal diagonals; equal diagonals alone do not make every quadrilateral a rectangle.
Revision checklist
- I can explain and apply what is a proposition?.
- I can explain and apply open sentences.
- I can explain and apply conditional statements.
- I can explain and apply converse.
- I can explain and apply counterexamples.
- I can explain and apply negation.
- I can explain and apply contrapositive.
- I can explain and apply inverse.
- I can explain and apply necessary and sufficient.
- I can explain and apply if and only if.
- I can explain and apply truth of an implication.
- I can explain and apply proof by direct reasoning.
- I can explain and apply proof by contrapositive.
- I can explain and apply proof by contradiction.
- I can explain and apply geometric converses.
- I can explain and apply quantifiers.
- I can explain and apply common mistakes.
Official textbook and accuracy note
Official NCERT textbook portal. Supplementary original study guide and practice exercises, not a verified reproduction of numbered NCERT exercises or official answer key.
Frequently asked questions
Explain what is a proposition? with an example.
A proposition is a declarative statement that has a definite truth value, true or false. 'Every square is a rectangle' is true.
Explain open sentences with an example.
A sentence with an unspecified variable may not yet have a fixed truth value. 'x>3' depends on the value or domain of x.
Explain conditional statements with an example.
'If P, then Q' is an implication, written P⇒Q. P is the hypothesis and Q the conclusion. If n is divisible by 4, then n is even.
Explain converse with an example.
The converse of P⇒Q is Q⇒P. The converse need not have the same truth value. If n is even, then n is divisible by 4 is false; n=6 disproves it.
Explain counterexamples with an example.
A universal claim is false if even one example satisfies its hypothesis but not its conclusion. 'All prime numbers are odd' is false because 2 is prime and even.
Explain negation with an example.
The negation of a statement P asserts that P is not true. Negation reverses truth value. Negation of 'all students passed' is 'at least one student did not pass'.
Explain contrapositive with an example.
The contrapositive of P⇒Q is not Q⇒not P; it is logically equivalent to the original. If divisible by 4 then even; if odd then not divisible by 4.
Explain inverse with an example.
The inverse of P⇒Q is not P⇒not Q; it need not be equivalent to the original. Not divisible by 4 does not imply odd: 6 is a counterexample.
Explain necessary and sufficient with an example.
P is sufficient for Q when P⇒Q; Q is necessary for P. If both directions hold, each is necessary and sufficient. Being divisible by 4 is sufficient for being even, not necessary.
Explain if and only if with an example.
P⇔Q means both P⇒Q and Q⇒P; this is a biconditional. An integer is even iff it is divisible by 2.
Explain truth of an implication with an example.
P⇒Q is false only when P is true and Q false; otherwise it is true in classical logic. 'If 2 is even, then 2 is odd' is false.
Explain proof by direct reasoning with an example.
Start from the hypothesis and use definitions or established facts to derive the conclusion. If n=2k, then n²=4k²=2(2k²), so n² is even.
Explain proof by contrapositive with an example.
Prove not Q⇒not P instead of P⇒Q when it is easier. If n² is odd, n is odd, equivalently if n is even, n² is even.
Explain proof by contradiction with an example.
Assume the claim false and derive an impossibility using established facts. A contradiction might be that an integer is simultaneously even and odd.
Explain geometric converses with an example.
Many geometry theorems have distinct converses requiring separate proof. If a quadrilateral is a rectangle it has equal diagonals; equal diagonals alone do not make every quadrilateral a rectangle.
