1. What is a Proposition?
Look at these sentences:
| Sentence | True, false, or neither? |
|---|---|
| \(5\) is a prime number. | True |
| \(7 + 3 = 11\). | False |
| Every even number is a multiple of \(4\). | False |
| What is your name? | Neither — it is a question |
| Please close the door. | Neither — it is a request |
Definition
A proposition is a sentence that can be judged to be either true or false (but not both at the same time). The truth value is its judgement: true or false. Questions, commands, wishes and exclamations are not propositions.
Four things matter for a proposition:
- It is a complete declarative sentence.
- It has a definite truth value true or false.
- The truth value does not change with the speaker.
- Both options ("true" or "false") are possible — a proposition simply takes one of them.
Example 1 — Spotting propositions
Question: Which of the following are propositions? (i) \(2\) is odd. (ii) How tall are you? (iii) Sit down. (iv) \(9\) is an even number. (v) A triangle has three sides.
Solution:
(i) Proposition (false). (ii) Not a proposition (question). (iii) Not a proposition (command). (iv) Proposition (false). (v) Proposition (true).
Propositions: (i), (iv) and (v).
2. Statements of the Type "If P, then Q"
Many propositions in mathematics are written as "If \(P\), then \(Q\)". Here \(P\) and \(Q\) are themselves statements.
Parts of an if-then statement
In "If \(P\), then \(Q\)", the statement \(P\) is called the hypothesis (also called the condition or antecedent) and the statement \(Q\) is called the conclusion (also called the consequent).
The truth of an if-then statement is decided by one question: whenever \(P\) is true, must \(Q\) also be true?
| Proposition (if-then form) | Hypothesis \(P\) | Conclusion \(Q\) | Truth value |
|---|---|---|---|
| If \(n\) is divisible by \(6\), then \(n\) is divisible by \(3\). | \(n\) is divisible by \(6\) | \(n\) is divisible by \(3\) | True |
| If \(n\) is odd, then \(n\) is prime. | \(n\) is odd | \(n\) is prime | False |
| If a figure is a square, then it is a rectangle. | the figure is a square | the figure is a rectangle | True |
Example 2 — Dividing a claim into P and Q
Question: Write "If a number ends with the digit 0, then it is divisible by 5" in the "If P, then Q" form and identify \(P\) and \(Q\).
Solution: \(P\): the number ends with the digit 0. \(Q\): the number is divisible by 5. The full statement is true.
\(P\) = "ends with 0", \(Q\) = "divisible by 5".
3. The Converse of a Proposition
Take an if-then proposition and swap its hypothesis and conclusion. The new proposition is called the converse.
Converse
The converse of "If \(P\), then \(Q\)" is "If \(Q\), then \(P\)".
Example 3 — Forming converses
Question: Write the converse of each proposition.
(i) If a number is divisible by \(6\), then it is divisible by \(3\).
(ii) If two angles of a triangle are equal, then their opposite sides are equal.
Solution:
(i) Converse: If a number is divisible by \(3\), then it is divisible by \(6\).
(ii) Converse: If two sides of a triangle are equal, then their opposite angles are equal.
4. A True Proposition May Have a False Converse
Here is the central warning of this chapter: if a proposition is true, its converse may still be false.
| Proposition (true) | Converse | Converse true or false? |
|---|---|---|
| If \(n\) is divisible by \(6\), then \(n\) is divisible by \(3\). | If \(n\) is divisible by \(3\), then \(n\) is divisible by \(6\). | False — e.g. \(9\) |
| If \(n\) ends with the digit \(5\), then \(n\) is divisible by \(5\). | If \(n\) is divisible by \(5\), then \(n\) ends with the digit \(5\). | False — e.g. \(10\) |
| If a figure is a square, then it is a rectangle. | If a figure is a rectangle, then it is a square. | False — a \(2 \times 3\) rectangle is not a square |
Common confusion
Just because a statement is true, its converse need not be. Every square is a rectangle, but not every rectangle is a square. The two are entirely different claims.
5. When Both the Proposition and Its Converse Are True
Sometimes both the proposition and its converse are true. Then the two parts are said to be equivalent, and we often write them in the combined form "\(P\) if and only if \(Q\)" (shortened to "iff").
| Proposition | Converse | Combined form |
|---|---|---|
| If a number is divisible by \(2\) and by \(3\), then it is divisible by \(6\). | If a number is divisible by \(6\), then it is divisible by \(2\) and by \(3\). | A number is divisible by \(6\) iff it is divisible by both \(2\) and \(3\). |
| If a triangle has three equal sides, then it has three equal angles. | If a triangle has three equal angles, then it has three equal sides. | A triangle is equilateral iff all its angles are equal. |
| If a number ends with the digit \(0\), then it is divisible by \(10\). | If a number is divisible by \(10\), then it ends with the digit \(0\). | A number is divisible by \(10\) iff it ends with \(0\). |
Exam tip
When the question says "show that \(P \iff Q\)", you must prove two separate things: that \(P\) implies \(Q\) and that \(Q\) implies \(P\). Missing the second one loses marks.
6. Counterexamples
How do we know a proposition is false? It is enough to find one single example where the hypothesis holds but the conclusion fails. Such an example is called a counterexample.
Counterexample
A counterexample to "If \(P\), then \(Q\)" is an instance in which \(P\) is true but \(Q\) is false. One genuine counterexample is enough to prove the proposition false — no matter how many other examples support it.
| Wrong proposition | Counterexample | Why it works |
|---|---|---|
| If \(n\) is odd, then \(n\) is prime. | \(n = 9\) | \(9\) is odd but \(9 = 3 \times 3\) is not prime. |
| Every even number is divisible by \(4\). | \(6\) | \(6\) is even but \(6 \div 4\) is not a whole number. |
| Every number ending in \(3\) is divisible by \(3\). | \(13\) | \(13\) ends in \(3\) but \(13 \div 3\) is not a whole number. |
Example 4 — Testing with a counterexample
Question: Is the proposition "If a number is divisible by \(4\), then it is divisible by \(8\)" true?
Solution: Test \(n = 12\). Here \(12\) is divisible by \(4\) (since \(12 = 4 \times 3\)), but \(12\) is not divisible by \(8\). So the hypothesis is true while the conclusion fails.
The proposition is false because \(n = 12\) is a counterexample.
7. Reasoning with Numbers: Multiples and Divisors
This chapter sharpens your language about numbers. A few proposition families reappear continually:
| Family | Example proposition | Truth value / note |
|---|---|---|
| Multiples of a divisor | Every multiple of \(6\) is a multiple of \(3\). | True (because \(6 = 2 \times 3\)) |
| Composites | Every even number is composite. | False — \(2\) is even and prime |
| Divisibility by product | If \(n\) is divisible by \(2\) and by \(5\), then \(n\) is divisible by \(10\). | True (since \(2\) and \(5\) have no common factor besides \(1\)) |
| Number of factors | If \(n\) is a perfect square, then \(n\) has an odd number of factors. | True — the next section |
Number-lore you can rely on
The sum of two even numbers is even. The sum of two odd numbers is even. The product of an even number and any whole number is even. And a number is divisible by \(2\) exactly when its last digit is even — these are all propositions whose converses are also carefully checked in this chapter.
8. Perfect Squares Have an Odd Number of Factors
Explore factors yourself:
| \(n\) | Is \(n\) a perfect square? | All factors of \(n\) | Number of factors |
|---|---|---|---|
| \(12\) | No | \(1, 2, 3, 4, 6, 12\) | \(6\) (even) |
| \(16\) | Yes (\(4^2\)) | \(1, 2, 4, 8, 16\) | \(5\) (odd) |
| \(18\) | No | \(1, 2, 3, 6, 9, 18\) | \(6\) (even) |
| \(25\) | Yes (\(5^2\)) | \(1, 5, 25\) | \(3\) (odd) |
| \(36\) | Yes (\(6^2\)) | \(1, 2, 3, 4, 6, 9, 12, 18, 36\) | \(9\) (odd) |
| \(49\) | Yes (\(7^2\)) | \(1, 7, 49\) | \(3\) (odd) |
| \(64\) | Yes (\(8^2\)) | \(1, 2, 4, 8, 16, 32, 64\) | \(7\) (odd) |
| \(81\) | Yes (\(9^2\)) | \(1, 3, 9, 27, 81\) | \(5\) (odd) |
Result
A natural number \(n\) has an odd number of factors iff \(n\) is a perfect square. Its converse is also true: if the number of factors is odd, then \(n\) is a perfect square.
Why? Group the factors of \(n\) in pairs whose product is \(n\):
| \(n\) | Pairs of factors \((d, n \div d)\) | The unpaired factor |
|---|---|---|
| \(12\) | \((1, 12), (2, 6), (3, 4)\) | none — all paired |
| \(16\) | \((1, 16), (2, 8)\) | \(4\) (since \(4 \times 4 = 16\)) |
| \(36\) | \((1, 36), (2, 18), (3, 12), (4, 9)\) | \(6\) (since \(6 \times 6 = 36\)) |
Every factor \(d\) of \(n\) can be paired with \(n \div d\). If \(n\) is not a perfect square, the two factors in every pair are different, so the pairs come out even. If \(n\) is a perfect square, its square root pairs with itself — one factor stays alone — so the total is odd.
9. Can You Disprove the Claim?
This section of the chapter gives you claims where a single counterexample destroys a hasty generalisation. Practise finding one.
Challenge 1
Claim: "If a number ends with the digit \(6\), then it is divisible by \(6\)."
Check: \(26\) ends with \(6\), but \(26 \div 6\) is not a whole number.
Counterexample: \(26\). The claim is false.
Challenge 2
Claim: "The sum of any two prime numbers is even."
Check: \(2 + 3 = 5\), which is odd.
Counterexample: \(2 + 3 = 5\). The claim is false.
Challenge 3
Claim: "If a number has exactly two factors, then it is odd."
Check: \(2\) has exactly two factors (\(1\) and \(2\)) and is even.
Counterexample: \(2\). The claim is false.
10. Geometric Propositions and Their Converses
Geometry is rich with if-then propositions. Here both directions are often (but not always) true.
| Proposition | Converse | Both true? |
|---|---|---|
| If two lines are parallel, then their alternate interior angles (cut by a transversal) are equal. | If alternate interior angles are equal, then the lines are parallel. | Yes |
| If a quadrilateral is a rectangle, then its diagonals are equal. | If the diagonals of a quadrilateral are equal, then it is a rectangle. | No (an isosceles trapezium can have equal diagonals) |
| If a triangle is isosceles, then two of its angles are equal. | If two angles of a triangle are equal, then two of its sides are equal. | Yes |
| If two triangles are congruent, then they have the same area. | If two triangles have the same area, then they are congruent. | No (a triangle and its doubled image may share base and height) |
Be careful with converse geometry claims
"Diagonal" statements are famous traps: equal diagonals do not force a rectangle, and equal sides do not force a square. Always check the converse separately.
11. The Baudhāyana (Pythagoras) Theorem as a Proposition
The rule relating the sides of a right-angled triangle is known in Indian tradition from the Baudhāyana’s Sulbasūtra (c. 800 BCE), where it was used when building altars. Textbooks today usually name it after the Greek mathematician Pythagoras.
Baudhāyana–Pythagoras theorem
In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides:
\[ \text{(hypotenuse)}^2 = \text{(first leg)}^2 + \text{(second leg)}^2 \]
Check with numbers:
| Triangle (legs, hypotenuse) | Check | Right-angled? |
|---|---|---|
| \(3, 4, 5\) | \(3^2 + 4^2 = 9 + 16 = 25 = 5^2\) | Yes |
| \(5, 12, 13\) | \(25 + 144 = 169 = 13^2\) | Yes |
| \(6, 8, 10\) | \(36 + 64 = 100 = 10^2\) | Yes |
| \(8, 15, 17\) | \(64 + 225 = 289 = 17^2\) | Yes |
| \(2, 3, 4\) | \(4 + 9 = 13 \neq 16\) | No |
| \(5, 6, 7\) | \(25 + 36 = 61 \neq 49\) | No |
Memory aid
The hypotenuse is the longest side, always opposite the right angle. Only ever square it on the left by itself.
12. The Converse of the Baudhāyana (Pythagoras) Theorem
Now form the converse: if the square on one side of a triangle equals the sum of the squares on the other two sides, is the triangle necessarily right-angled? Here — unusually — the converse is true.
Converse of the Baudhāyana–Pythagoras theorem
If, in a triangle with longest side \(c\), we have \(a^2 + b^2 = c^2\), then the angle opposite \(c\) is a right angle.
Example 5 — Using the converse
Question: A triangle has sides \(9\) cm, \(12\) cm and \(15\) cm. Is it right-angled? If so, which vertex?
Solution: The longest side is \(15\). Check \(9^2 + 12^2 = 81 + 144 = 225 = 15^2\). The equality holds, so a right angle is opposite the \(15\) cm side.
Yes — right-angled, with the right angle opposite the \(15\) cm side.
Challenge 4 — counterexample to the careless converse
Claim: "If a triangle has a right angle, then its side lengths form consecutive numbers."
Check: \(6, 8, 10\) form a right triangle (\(36 + 64 = 100\)) but are not consecutive.
Counterexample: \(6, 8, 10\). The claim is false.
13. Worked Examples
Follow the four part pattern used throughout this chapter: Given → Required → Reasoning → Solution.
Worked Example 1 — Given / Required / Reasoning / Solution
Given: The proposition "If \(n\) is a multiple of \(5\), then \(n\) ends with \(5\)".
Required: Decide whether it is true; if false, give a counterexample.
Reasoning: A multiple of \(5\) may end with \(5\) or with \(0\) (\(10, 20, 30, \ldots\)). So the conclusion does not always follow.
Solution: \(n = 10\) is a multiple of \(5\) but ends with \(0\). The proposition is false.
False — counterexample \(10\).
Worked Example 2 — Given / Required / Reasoning / Solution
Given: A triangle with sides \(7\) cm, \(24\) cm and \(25\) cm.
Required: Check whether it is right-angled using the converse of the Baudhāyana theorem.
Reasoning: \(25\) is the longest side; compute \(7^2 + 24^2\).
Solution: \(49 + 576 = 625 = 25^2\). The equality holds.
Right-angled; right angle opposite the \(25\) cm side.
Worked Example 3 — Given / Required / Reasoning / Solution
Given: The claim "Every prime number is odd".
Required: Disprove it if possible.
Reasoning: A single counterexample is enough.
Solution: \(2\) is prime, yet even.
Claim false — counterexample \(2\).
14. Fermat’s Claim and Euler’s Counterexample
A powerful historical episode shows exactly why counterexamples matter. The mathematician Fermat looked at these numbers — each one is "2 raised to a power of two, plus 1":
| \(n\) | \(2^{2^n} + 1\) | Value | Prime or not? |
|---|---|---|---|
| \(0\) | \(2^{2^0} + 1\) | \(2^1 + 1 = 3\) | Prime |
| \(1\) | \(2^{2^1} + 1\) | \(2^2 + 1 = 5\) | Prime |
| \(2\) | \(2^{2^2} + 1\) | \(2^4 + 1 = 17\) | Prime |
| \(3\) | \(2^{2^3} + 1\) | \(2^8 + 1 = 257\) | Prime |
| \(4\) | \(2^{2^4} + 1\) | \(2^{16} + 1 = 65537\) | Prime |
| \(5\) | \(2^{2^5} + 1\) | \(2^{32} + 1 = 4294967297\) | Not prime — see below |
Fermat observed the pattern for \(n = 0\) to \(4\) and claimed the numbers \(2^{2^n} + 1\) are always prime. About a century later, Euler found one example defeating the claim:
Euler’s counterexample
\[ 2^{32} + 1 = 4294967297 = 641 \times 6700417 \]
So the proposition "\(2^{2^n} + 1\) is prime for every whole number \(n\)" is false, and a single number killed the whole claim. Four or five confirming examples were never enough — one counterexample was.
15. Comparison: Proposition, Converse, Counterexample
| Idea | What it is | How it is used | Example |
|---|---|---|---|
| Proposition | A sentence that is true or false (not both). | Stated, then tested. | "If \(n\) is divisible by \(6\), then \(n\) is divisible by \(3\)." |
| Converse | Same proposition with hypothesis and conclusion swapped. | Checked separately — truth does not pass automatically. | "If \(n\) is divisible by \(3\), then \(n\) is divisible by \(6\)." |
| Counterexample | One instance where \(P\) holds but \(Q\) fails. | Disproves the proposition decisively. | \(9\): divisible by \(3\), but not by \(6\). |
16. Think and Reflect
- If a statement is true for the first fifty numbers you test, are you certain it is true for all numbers? The Fermat numbers teach a different lesson.
- Why is one counterexample more powerful for disproving than a hundred examples are for proving?
- The converse of a true proposition is frequently false. Name a proposition you have used and check its converse.
- For which class of numbers is the number of factors odd? Can you give the converse as well?
- Why must you identify the longest side before applying the converse of the Baudhāyana theorem?
17. Common Mistakes
| # | Mistake | How to avoid it |
|---|---|---|
| 1 | Assuming the converse of a true proposition is true. | Write the converse separately and test it with a counterexample. |
| 2 | Confusing a question or command with a proposition. | Propositions are declarative sentences with a definite truth value. |
| 3 | Labelling an example that does not satisfy the hypothesis as a counterexample. | A counterexample must make the hypothesis true and the conclusion false. |
| 4 | Squaring the wrong side when using the converse of the Baudhāyana theorem. | Always take the longest side as the candidate hypotenuse. |
| 5 | Believing four examples prove a general claim. | Proof covers all cases; one counterexample disproves. |
| 6 | Quoting the Baudhāyana–Pythagoras theorem with the hypotenuse added, not squared. | State it as \(\text{(hypotenuse)}^2 = a^2 + b^2\) before substituting. |
18. Quick Revision — Remember These
Proposition
A sentence that is true or false (never both). Questions, commands and wishes are not propositions.
If-then form
"If \(P\), then \(Q\)": \(P\) is the hypothesis, \(Q\) the conclusion.
Converse
Swap \(P\) and \(Q\). Truth does not carry over automatically.
Counterexample
One instance with \(P\) true and \(Q\) false disproves the proposition.
Iff
When proposition and converse are both true: \(P \iff Q\).
Perfect squares
\(n\) is a perfect square iff it has an odd number of factors.
Fermat numbers
\(2^{2^n} + 1\): prime for \(n=0\) to \(4\), composite for \(n=5\) (Euler).
Baudhāyana theorem
\(c^2 = a^2 + b^2\) in a right triangle, \(c\) the hypotenuse.
Converse theorem
If \(a^2 + b^2 = c^2\) (longest side \(c\)), the angle opposite \(c\) is right.
Favourite triples
\(3{-}4{-}5\), \(5{-}12{-}13\), \(6{-}8{-}10\), \(8{-}15{-}17\).
19. Chapter Summary
What you should be able to do
1. Recognise propositions and reject questions/commands/wishes.
2. Split "If \(P\), then \(Q\)" into hypothesis and conclusion.
3. Write the converse and test it independently.
4. Prove a proposition false with one counterexample.
5. Justify why a perfect square has an odd number of factors.
6. State the Baudhāyana–Pythagoras theorem and its converse, and apply them to triangles.
20. Self-Practice Questions
Section A — Multiple Choice Questions
- Which of the following is not a proposition? (a) \(7\) is even. (b) Please sit down. (c) \(3 + 5 = 8\). (d) Monday is a day.
- The converse of "If a number is divisible by \(10\), then it ends with \(0\)" is: (a) If a number ends with \(0\), then it is divisible by \(10\). (b) If it does not end with \(0\), it is not divisible by \(10\). (c) If a number is divisible by \(10\), then it is even. (d) 10 is divisible by the number.
- Which number is a counterexample to "If \(n\) is odd, then \(n\) is prime"? (a) \(3\) (b) \(7\) (c) \(9\) (d) \(13\)
- The proposition "If \(n\) is divisible by \(4\), then \(n\) is divisible by \(8\)" is: (a) true and its converse is true (b) false (c) true (d) not a proposition
- How many factors does \(36\) have? (a) \(7\) (b) \(8\) (c) \(9\) (d) \(10\)
- Which number has an odd number of factors? (a) \(18\) (b) \(24\) (c) \(25\) (d) \(50\)
- The first Fermat number \(2^{2^n} + 1\) that is not prime is: (a) \(3\) (b) \(17\) (c) \(257\) (d) \(2^{32} + 1\)
- A triangle has sides \(6\) cm, \(8\) cm, \(10\) cm. The angle opposite the \(10\) cm side is: (a) \(60^\circ\) (b) an acute angle (c) a right angle (d) not determined
- Which set of sides is a Pythagorean triple? (a) \(2, 3, 4\) (b) \(5, 12, 13\) (c) \(5, 6, 7\) (d) \(4, 5, 6\)
- The statement "\(2 + 2 = 5\)" is: (a) a question (b) a command (c) a false proposition (d) a true proposition
- The converse of the Baudhāyana theorem says: if \(a^2 + b^2 = c^2\) with \(c\) the longest side, then the triangle is (a) equilateral (b) right-angled (c) obtuse-angled (d) isosceles
- \(2^{32} + 1 = 641 \times 6700417\) shows that: (a) \(2^5 + 1\) is prime (b) Fermat’s claim fails for \(n = 5\) (c) \(641\) is even (d) every Fermat number is composite
Section B — Very Short Answer Questions
- State whether "Is \(9\) odd?" is a proposition.
- For "If \(n\) is divisible by \(6\), then \(n\) is divisible by \(3\)", name the hypothesis and conclusion.
- Write the converse of "If a number ends with \(0\), then it is divisible by \(5\)."
- Give a counterexample to "Every odd number is prime."
- How many factors does \(49\) have?
- Check whether \(5\) cm, \(12\) cm, \(13\) cm form a right-angled triangle.
- Write "A square is a rectangle" in if-then form.
- What is the value of \(2^{2^2} + 1\)?
Section C — Short Answer Questions
- Which of "What time is it?", "\(4\) is prime", "Feed the cat", "\(6\) has four factors" are propositions? Give reasons.
- Write the converse of "If a number is divisible by \(6\), then it is even." Is the converse true? Give a counterexample if it is false.
- A triangle has sides \(9\) cm, \(12\) cm, \(15\) cm. Using the converse of the Baudhāyana theorem, decide if it is right-angled. Which angle is the right angle?
- Give two examples of a true proposition whose converse is false, with a counterexample for each.
- Explain, in words, why \(36\) has an odd number of factors while \(12\) has an even number.
- State the proposition "If \(n\) is even, then \(n\) is divisible by \(2\)" and its converse. Are both true?
- Show that \(2, 3, 4\) are not the side lengths of a right-angled triangle.
- Fermat’s numbers for \(n=0\) to \(4\) are prime. Why does that not prove the claim for all \(n\)?
Section D — Reasoning Questions
- Divya says, "\(6\) is divisible by \(3\), and \(9\) is divisible by \(3\), so every number divisible by \(3\) is divisible by \(6\)." Where is the flaw in her reasoning?
- Can a command be a proposition? Explain why or why not.
- Suppose a proposition and its converse are both true. What can you conclude? Give an example.
- A number ends with the digit \(0\). Without dividing, explain whether it is divisible by \(10\) and by \(2\). Justify.
- Why must a counterexample make the hypothesis true? Use "If \(n\) is odd, then \(n\) is prime" with \(n = 4\) to explain.
- Ravi claims a triangle with sides \(2, 4, 6\) is right-angled because \(2^2 + 4^2 = 4 + 16 = 20\) and \(6^2 = 36\). Why is he wrong?
- A figure has four equal sides. Can you conclude it is a square? Explain with the converse idea.
- Order the following propositions by "most likely to be worth checking its converse": all primes are odd; every multiple of \(6\) is a multiple of \(3\); a rectangle’s diagonals are equal. Justify briefly.
Section E — Assertion–Reason Questions
- A: A perfect square has an odd number of factors. R: A perfect square is of the form \(m^2\), and every other factor pairs with a different factor except \(m\) itself.
- A: The converse of a true proposition is always true. R: The converse is formed by interchanging the hypothesis and conclusion.
- A: \(9\) is a counterexample to "If \(n\) is odd, then \(n\) is prime." R: \(9\) is odd and \(9 = 3 \times 3\).
- A: A triangle with sides \(6, 8, 10\) is right-angled. R: \(6^2 + 8^2 = 100 = 10^2\).
- A: \(2^{32} + 1\) is a counterexample to a claim by Fermat. R: \(2^{32} + 1 = 641 \times 6700417\).
Section F — Case-Based Questions
Case 1 — A carpenter’s right angle
A carpenter checks a table frame. She knows that if the frame is a rectangle, both diagonals are equal. She measures the two diagonals and finds them equal.
(a) Which proposition is she using? (b) Does equality of diagonals prove the frame is a rectangle? (c) Give a counterexample showing why the converse can fail.
(a) "If a frame is a rectangle, then its diagonals are equal." (b) No — equality of diagonals is the converse. (c) An isosceles trapezium frame can have equal diagonals without being a rectangle.
Case 2 — Fencing a right corner
A gardener makes a corner where two fence lines meet. He wants them perpendicular. He marks \(3\) m along one line and \(4\) m along the other, and measures \(5\) m between the two marks.
(a) Which theorem is he applying? (b) Justify that a right angle is unavoidable. (c) If he used \(6\) m and \(8\) m instead, what length should he measure between the marks?
(a) The converse of the Baudhāyana–Pythagoras theorem. (b) \(3^2 + 4^2 = 5^2\), so the angle opposite \(5\) is right. (c) \(10\) m, because \(6^2 + 8^2 = 10^2\).
Case 3 — A student’s claim
Meera tests the proposition "If a number is divisible by \(6\), then it is divisible by \(12\)". She checks \(12\), \(24\), \(36\), \(48\) and concludes it is true.
(a) Is her conclusion justified? (b) If not, give a counterexample. (c) State the converse and test whether it is true.
(a) No — examples do not prove. (b) \(6\) is divisible by \(6\) but not by \(12\). (c) Converse: "If divisible by \(12\), then divisible by \(6\)" — true.
Section G — HOTS Questions
- Is "This sentence is false" a proposition? Give a careful reason.
- A number has exactly two factors. Write the proposition "If a number has exactly two factors, then it is prime" and its converse. Are both true?
- A triangle has sides \(7, 24, 25\). Show it is right-angled, then write the converse proposition that applies.
- Prove that the number of factors of \(100\) is odd without listing all factors. (Hint: \(100 = (2^2)(5^2)\) and factor pairs.)
- Design a true proposition of your own about two-digit numbers whose converse is false, and give its counterexample.
21. Answers
Section A — MCQs: (1) b (2) a (3) c (4) b (5) c (6) c (7) d (8) c (9) b (10) c (11) b (12) b
Section B — VSA:
- No — it is a question.
- Hypothesis: \(n\) divisible by \(6\). Conclusion: \(n\) divisible by \(3\).
- If a number is divisible by \(5\), then it ends with \(0\). (False — \(5\) ends with \(5\).)
- \(9\) (or \(15, 21, \ldots\)).
- \(3\) (factors \(1, 7, 49\)).
- Yes — \(25 + 144 = 169 = 13^2\).
- If a figure is a square, then it is a rectangle.
- \(2^{4} + 1 = 17\).
Section C — SA:
- Propositions: "\(4\) is prime" (false) and "\(6\) has four factors" (true). The question and the command are not.
- Converse: "If a number is even, then it is divisible by \(6\)." False — \(8\) is even but not divisible by \(6\).
- \(9^2 + 12^2 = 81 + 144 = 225 = 15^2\), so it is right-angled; right angle opposite \(15\) cm.
- (i) "If \(n\) is divisible by \(6\), then \(n\) is divisible by \(3\)" — converse false, counterexample \(9\). (ii) "If a figure is a square, then it is a rectangle" — converse false, a \(2\times 3\) rectangle.
- Factors pair as \((d, 36/d)\). For \(36 = 6^2\), the pair \((6, 6)\) is one factor counted once, so the total is odd. For \(12\), every factor \(d\) has a different partner \(12/d\).
- Proposition: If \(n\) is even, then \(n\) is divisible by \(2\). Converse: If \(n\) is divisible by \(2\), then \(n\) is even. Both true.
- Longest side \(4\): \(2^2 + 3^2 = 4 + 9 = 13 \neq 16 = 4^2\).
- Examples cannot prove a claim about all \(n\). The claim fails at \(n = 5\): \(2^{32} + 1 = 641 \times 6700417\).
Section D — Reasoning:
- Her conclusion is the converse of the true statement "divisible by 6 implies divisible by 3", and she tested only multiples of \(6\). A counterexample like \(9\) shows the converse is false.
- No — a command gives instructions; it has no truth value.
- They are equivalent (\(P \iff Q\)). Example: "divisible by \(2\) and \(3\)" iff "divisible by \(6\)".
- It ends with \(0\), so it is divisible by \(10\) and also by \(2\) (since \(2\) divides \(10\) and \(0\) is even).
- To disprove "If \(P\) then \(Q\)" you need \(P\) true and \(Q\) false. With \(n=4\), the hypothesis "\(4\) is odd" is false, so it is irrelevant to the proposition.
- \(2 + 4 = 6\) means the sides \(2, 4, 6\) cannot even form a triangle (degenerate), and \(4 + 16 = 20 \neq 36\).
- Four equal sides force a rhombus, not a square. "Equal sides" is the converse part and does not imply right angles.
- "All primes are odd" — easiest to disprove (counterexample \(2\)); "every multiple of \(6\) is a multiple of \(3\)" — true, converse false; "diagonals of a rectangle are equal" — converse false. Checking converses matters most where the claim is narrow, like the rectangle diagonal statement.
Section E — Assertion–Reason: (1) Both true, R explains A. (2) A false, R true. (3) Both true, R explains A. (4) Both true, R explains A. (5) Both true, R explains A.
Section F — Case-based: Answers are given under each case.
Section G — HOTS:
- It has no stable truth value — if true it is false, if false it is true — so it is not a proposition (a paradox).
- Converse: "If a number is prime, then it has exactly two factors." Both are true: a number with exactly two factors (1 and itself) is prime, and a prime has exactly those two factors.
- \(7^2 + 24^2 = 49 + 576 = 625 = 25^2\), so the triangle is right-angled. Converse: if \(a^2 + b^2 = c^2\) (longest \(c\)), the angle opposite \(c\) is a right angle.
- \(100 = 2^2 \times 5^2\). Every factor is \(2^a \times 5^b\) with \(a \in \{0,1,2\}\), \(b \in \{0,1,2\}\), giving \(3 \times 3 = 9\) factors — odd.
- Example: "If a two-digit number is a square, then it is divisible by \(3\)." Counterexample: \(16 = 4^2\) is a square but is not divisible by \(3\).