1. Chapter at a Glance
"If it rains, the ground gets wet." True. But "If the ground is wet, it has rained"? Not necessarily — someone may have watered the garden. Swapping the "if" part and the "then" part of a statement gives its converse, and the converse may or may not be true. Mathematics is full of such if–then statements, especially in geometry and number theory. This chapter teaches you to state propositions precisely, form their converses, and decide — by proof or by counterexample — which ones are true.
- Propositions (mathematical statements) and their truth values.
- Conditional statements "If P, then Q": hypothesis and conclusion.
- Converse, inverse and contrapositive.
- Counterexamples — one is enough to disprove a general claim.
- "If and only if" (biconditional) statements.
- Theorems and their converses in geometry and number theory.
- Methods of proof: direct proof, proof by contradiction.
2. Propositions
A proposition (or mathematical statement) is a sentence that is either definitely true or definitely false — not both.
| Sentence | Proposition? | Truth value |
|---|---|---|
| 7 is a prime number. | Yes | True |
| The sum of angles of a triangle is 200°. | Yes | False |
| Close the door. | No (a command) | — |
| Mathematics is fun. | No (an opinion) | — |
| x + 2 = 5 | Not until x is known | Depends on x |
The negation of P ("not P") has the opposite truth value. Negation of "All primes are odd" is "Some prime is not odd" (true — 2).
3. If–Then Statements
A conditional has the form "If P, then Q" (written P ⇒ Q). P is the hypothesis (given part); Q is the conclusion.
Many statements hide an if–then form: "The diagonals of a rectangle are equal" means "If a quadrilateral is a rectangle, then its diagonals are equal".
P ⇒ Q is false only when P is true and Q is false.
4. Converse, Inverse, Contrapositive
| Form | Statement | Example (P: n is divisible by 6; Q: n is divisible by 3) | True? |
|---|---|---|---|
| Original | P ⇒ Q | If n is divisible by 6, then n is divisible by 3. | True |
| Converse | Q ⇒ P | If n is divisible by 3, then n is divisible by 6. | False (n = 9) |
| Inverse | not P ⇒ not Q | If n is not divisible by 6, then n is not divisible by 3. | False (n = 9) |
| Contrapositive | not Q ⇒ not P | If n is not divisible by 3, then n is not divisible by 6. | True |
- A statement and its contrapositive are always both true or both false.
- The converse (and inverse) may be true or false — it must be checked separately.
5. Counterexamples
To show a general statement ("for all …", "every …", "if … then …") is false, one example where the hypothesis holds but the conclusion fails is enough. That example is a counterexample.
Claim: "If n is a prime, then n is odd." Is it true?
Solution: n = 2 is prime but even. The claim is false.
Claim: "n² + n + 41 is prime for every whole number n." Test.
Solution: It works for n = 0 to 39, but n = 40 gives 1681 = 41². False. Many examples never prove a general claim — only a proof does.
6. "If and Only If"
When both P ⇒ Q and Q ⇒ P are true, we write P ⇔ Q, read "P if and only if Q". Then P and Q are equivalent conditions.
| Statement | Converse true? | Combined |
|---|---|---|
| If a triangle is right-angled, a² + b² = c². | Yes | Right-angled ⇔ a² + b² = c² |
| If two sides of a triangle are equal, the angles opposite them are equal. | Yes | Equal sides ⇔ equal opposite angles |
| If a quadrilateral is a parallelogram, its diagonals bisect each other. | Yes | Parallelogram ⇔ diagonals bisect each other |
| If a quadrilateral is a square, its diagonals are equal. | No (rectangle) | Only one direction |
| If two angles are vertically opposite, they are equal. | No | Only one direction |
| If n is even, n² is even. | Yes | n even ⇔ n² even |
7. Methods of Proof
Direct proof
Prove: If n is odd, then n² is odd.
Proof: n odd → n = 2k + 1. n² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is odd. ∎
Proof using the contrapositive
Prove the converse: If n² is even, then n is even.
Proof: Contrapositive: "If n is odd, then n² is odd" — just proved in Example 3. A statement is true exactly when its contrapositive is, so the converse holds. ∎
Proof by contradiction
Assume the conclusion is false and show this leads to something impossible. (Used to prove √2 is irrational.)
Prove: There is no largest even number.
Proof: Suppose N is the largest even number. Then N + 2 is even and bigger than N — contradiction. So no largest even number exists. ∎
8. Theorem and Converse in Geometry
Theorem: If ∠C = 90° in △ABC, then AB² = BC² + CA².
Converse: If AB² = BC² + CA², then ∠C = 90°.
Proof of converse: Construct △PQR with ∠R = 90°, QR = BC and RP = CA. By the theorem, PQ² = BC² + CA² = AB², so PQ = AB. Then △ABC ≅ △PQR (SSS), so ∠C = ∠R = 90°. ∎
Theorem: If AB = AC, then ∠B = ∠C. (Draw the bisector of ∠A; SAS congruence.)
Converse: If ∠B = ∠C, then AB = AC. (Draw the bisector of ∠A; AAS congruence.)
A false converse: "If two triangles are congruent, their areas are equal" is true, but "If two triangles have equal areas, they are congruent" is false — a 4 × 3 right triangle and a triangle with base 6 and height 2 both have area 6.
9. Solved Problems (Board Pattern)
Write the converse of "If a number ends in 0, it is divisible by 5" and decide whether it is true.
Solution: Converse: "If a number is divisible by 5, it ends in 0." False — 15 is divisible by 5 but ends in 5.
Write the contrapositive of "If two lines are parallel, they do not meet."
Solution: "If two lines meet, they are not parallel." (True, like the original.)
Prove that the sum of two consecutive odd numbers is divisible by 4.
Solution: (2k + 1) + (2k + 3) = 4k + 4 = 4(k + 1). ∎
10. Master Formula Table
| Form | Symbol | Same truth as original? |
|---|---|---|
| Conditional | P ⇒ Q | — |
| Converse | Q ⇒ P | Not necessarily |
| Inverse | ¬P ⇒ ¬Q | Not necessarily (same as converse) |
| Contrapositive | ¬Q ⇒ ¬P | Always |
| Biconditional | P ⇔ Q | Both P ⇒ Q and Q ⇒ P true |
| Even / odd | 2k / 2k + 1 | Used in number proofs |
11. Master Summary Table
| Idea | Key point | Exam focus |
|---|---|---|
| Proposition | True or false, not both | Identify statements |
| Converse | Swap hypothesis and conclusion | Write and test converses |
| Contrapositive | Equivalent to original | Indirect proofs |
| Counterexample | One case disproves | Disprove claims |
| If and only if | Both directions true | Pythagoras, isosceles, parallelogram |
| Proof methods | Direct, contradiction, contrapositive | Short number proofs |
12. Mind Map
- True/false
- Negation
- Hypothesis, conclusion
- Converse, contrapositive
- Counterexample
- Proof
- Pythagoras ⇔
- Divisibility
13. Tips and Tricks
- Rewrite any statement in "If …, then …" form before forming the converse.
- To test a converse, look for special cases: 2 (the even prime), squares vs rectangles, 0 and 1.
- If proving Q ⇒ P directly is hard, prove "not P ⇒ not Q" instead.
- Assuming the converse of a true theorem is automatically true.
- Thinking many examples prove a general statement.
- Negating "all are" as "none are" — the correct negation is "some are not".
14. Important One-liners
- A proposition is either true or false.
- The converse of P ⇒ Q is Q ⇒ P.
- A statement and its contrapositive are logically equivalent.
- One counterexample disproves a general claim.
- The converse of the Baudhāyana–Pythagoras theorem is true.
- "Every square is a rectangle" is true; its converse is false.
15. Exam-pattern Questions with Detailed Answers
Which is a proposition?
- Please sit down.
- Is 5 odd?
- Every rhombus is a square.
- What a lovely day!
Show answer
(C) is a statement with a definite truth value (false).
Answer: (C)
Write the converse of "If a triangle is equilateral, then it is isosceles."
Show answer
"If a triangle is isosceles, then it is equilateral." This is false (sides 5, 5, 8).
Answer: As above; false
Give a counterexample: "The square of any number is greater than the number."
Show answer
0.5² = 0.25 < 0.5 (also 1² = 1, 0² = 0).
Answer: e.g. 0.5
Write the negation of "All quadrilaterals have four right angles."
Show answer
"Some quadrilateral does not have four right angles." (True.)
Answer: As above
Write the contrapositive of "If x² = 9, then x = 3." Is the original true?
Show answer
Contrapositive: "If x ≠ 3, then x² ≠ 9." Both are false: x = −3 is a counterexample.
Answer: Both false (x = −3)
Is the converse of "If a quadrilateral is a rhombus, its diagonals are perpendicular" true?
Show answer
Converse: "If the diagonals are perpendicular, it is a rhombus." False — a kite has perpendicular diagonals but is not always a rhombus.
Answer: No (counterexample: kite)
Prove that the product of two odd numbers is odd.
Show answer
(2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1, odd. ∎
Answer: Proof as shown
State the theorem and its converse combined as an "if and only if" statement: "If a triangle has sides 3k, 4k, 5k, it is right-angled."
Show answer
The converse "If a triangle is right-angled, its sides are 3k, 4k, 5k" is false (5, 12, 13). So no "if and only if" statement is possible.
Answer: Converse false; cannot combine
Prove: If n² is divisible by 3, then n is divisible by 3.
Show answer
Contrapositive: if n is not divisible by 3, n = 3k ± 1. Then n² = 9k² ± 6k + 1 = 3(3k² ± 2k) + 1, not divisible by 3. Hence the original holds. ∎
Answer: Proof via contrapositive
Prove the converse of the isosceles triangle theorem: in △ABC, if ∠B = ∠C then AB = AC.
Show answer
Draw the bisector AD of ∠A meeting BC at D. In △ABD and △ACD: ∠BAD = ∠CAD, ∠B = ∠C, AD common. By AAS, the triangles are congruent, so AB = AC. ∎
Answer: Proof as shown
Claim: "If a + b is even, then both a and b are even." Find a counterexample, then correct the statement.
Show answer
a = 3, b = 5: sum 8 even, both odd. Correct version: "If a + b is even, then a and b are both even or both odd."
Answer: Counterexample 3, 5; corrected as above
Prove by contradiction that there are infinitely many prime numbers.
Show answer
Suppose there are only finitely many primes p₁, p₂, …, pₖ. Let N = p₁p₂…pₖ + 1. N is greater than 1, so it has a prime factor. But dividing N by any pᵢ leaves remainder 1, so none of them divides N — a new prime exists. Contradiction. ∎ (This argument is due to Euclid.)
Answer: Proof as shown
16. Quick Revision Summary
- A proposition is true or false; its negation flips the truth value.
- P ⇒ Q; converse Q ⇒ P; contrapositive ¬Q ⇒ ¬P (always equivalent).
- A converse must be proved separately; one counterexample disproves.
- When both directions hold, write P ⇔ Q.
- Proof methods: direct, contrapositive, contradiction.