Chapter 9 — Propositions and their Converses

Suppose a friend tells you, ‘Every number that ends in 5 is divisible by 5.’ Another says, ‘Please pass the salt.’ A third asks, ‘Is 91 a prime number?’ Only the first sentence can be checked and judged as right or wrong. The second is a request and the third is a question; neither of them is true or false.

Definition. A proposition is a statement that is either true or false, but not both. Mathematics is built out of propositions: every theorem you have learnt, such as ‘the angles of a triangle add up to 180°’, is a proposition that has been shown to be true.

Which sentences are propositions?

The ‘if … then …’ form. Most results in mathematics have two parts: a condition and a result that follows from it. We write them as ‘if X, then Y’. The part X is called the hypothesis (what is given) and Y is the conclusion (what follows). For example, in ‘if a number is a multiple of 10, then it ends in 0’, the hypothesis is ‘the number is a multiple of 10’ and the conclusion is ‘it ends in 0’.

Other ways of saying the same thing. The same proposition can be written in several ways, and you must learn to spot the hypothesis in each:

• ‘X implies Y’: being a multiple of 10 implies ending in 0.
• ‘Y when X’ or ‘Y whenever X’: a number ends in 0 whenever it is a multiple of 10.
• ‘Every X is Y’ or ‘All X are Y’: every multiple of 10 ends in 0.

All four forms carry exactly the same meaning. When you meet a sentence like ‘the base angles of an isosceles triangle are equal’, rewrite it as ‘if a triangle is isosceles, then its base angles are equal’. This makes it clear what you are allowed to assume and what you must show.

Why this matters. Once a statement is in ‘if X, then Y’ form, two important questions open up. Is the statement true, and how do we prove it? And what happens if we turn it around? The rest of this chapter is about these two questions.

In an earlier class you proved: if two sides of a triangle are equal, then the angles opposite them are equal. Now read this one: if two angles of a triangle are equal, then the sides opposite them are equal. The two sentences use the same ingredients, but the roles have been swapped. What was given in the first is to be proved in the second.

Definition. The converse of the proposition ‘if X, then Y’ is the proposition ‘if Y, then X’. We get it by exchanging the hypothesis and the conclusion.

Examples of converses.

Keeping the setting fixed. Many statements begin with a sentence that only sets the scene, such as ‘Let ABC be a triangle’ or ‘Let n be a positive whole number’. This sentence is not part of the hypothesis that gets swapped. When you form the converse, keep the setting sentence as it is and exchange only the ‘if’ part and the ‘then’ part. For example, take: Let a, b, c be the sides of a triangle, with c the longest. If the triangle has a right angle, then a² + b² = c². Its converse is: Let a, b, c be the sides of a triangle, with c the longest. If a² + b² = c², then the triangle has a right angle.

Converse of a statement in another form. First rewrite the statement in if-then form, then swap. ‘Every rhombus has perpendicular diagonals’ becomes ‘if a quadrilateral is a rhombus, then its diagonals are perpendicular’. Its converse is ‘if the diagonals of a quadrilateral are perpendicular, then it is a rhombus’. (Is that true? Think of a kite.)

The big question. If a proposition is true, must its converse be true? Look again at the table. The first proposition is true, but 9 is a multiple of 3 and not of 6, so its converse is false. The rain example shows the same thing in daily life: the road may be wet because a water tanker passed by. So the answer is no. The truth of a proposition tells us nothing, by itself, about the truth of its converse. Each one must be checked separately, either by a proof or by a counterexample, which is the subject of the next topic.

Someone claims, ‘All birds can fly.’ You do not need to examine every bird in the world to show that this is wrong. A single penguin is enough. An example that shows a general statement is false is called a counterexample.

Why one is enough. A statement like ‘if X, then Y’ claims that Y follows every time X happens. So one case in which X is true but Y is false destroys the claim. On the other hand, a hundred examples that agree with the statement do not prove it. There may be a case further along that breaks it. This is why mathematicians ask for a proof for ‘true’ and a counterexample for ‘false’.

A famous story. Around 1640, Pierre de Fermat noticed that the numbers 2¹ + 1 = 3, 2² + 1 = 5, 2⁴ + 1 = 17, 2⁸ + 1 = 257 and 2¹⁶ + 1 = 65537 are all prime. These are the numbers 2⁽²ⁿ⁾ + 1 for n = 0, 1, 2, 3, 4. He believed all such numbers are prime. About a hundred years later, Leonhard Euler took the next one, 2³² + 1 = 4294967297, and showed that it equals 641 × 6700417. One counterexample settled the question.

Formulas that ‘seem’ to give primes. Check these yourself:

How to look for a counterexample.

• Write the statement in if-then form.
• Look for a case where the hypothesis holds. Try small numbers, special numbers (0, 1, 2, negative numbers, squares) and special shapes (squares, very thin triangles).
• Check whether the conclusion fails. If it does, you have a counterexample; state it clearly with the numbers.

A good counterexample is specific. ‘Some multiples of 3 are not multiples of 6’ is vague. ‘9 is a multiple of 3 but not of 6’ is a counterexample. Notice too that a counterexample must satisfy the hypothesis. To disprove ‘if n is prime, then n is odd’, the number 9 is useless (it is not prime); the number 2 is the counterexample.

A shopkeeper says, ‘If you pay ₹ 500, then you get the bag.’ That is true. But it does not follow that everyone carrying the bag paid ₹ 500; one customer may have received it as a gift. This pattern, true one way and false the other way, is very common in mathematics.

Example 1: multiples. Proposition: if a number is a multiple of 6, then it is a multiple of 3. Proof: a multiple of 6 can be written 6k = 3 × (2k), which is 3 times a whole number. Converse: if a number is a multiple of 3, then it is a multiple of 6. Counterexample: 15 = 3 × 5 is a multiple of 3, but 15 ÷ 6 = 2.5. So the converse is false.

Example 2: area and congruence. Proposition: if two triangles are congruent, then they have the same area. This is true, because one fits exactly on the other and covers the same region. Converse: if two triangles have the same area, then they are congruent. Counterexample: a triangle with base 6 cm and height 2 cm, and another with base 4 cm and height 3 cm. Each has area ½ × 6 × 2 = ½ × 4 × 3 = 6 cm², but their bases differ, so they cannot be placed one on the other.

Example 3: squares. Proposition: if a quadrilateral is a square, then all its angles are equal. True, since every angle of a square is 90°. Converse: if all the angles of a quadrilateral are equal, then it is a square. Counterexample: a rectangle 5 cm by 3 cm has four 90° angles but unequal sides.

Example 4: divisibility. Proposition: if n is divisible by 24, then n is divisible by both 4 and 6. True, because 24 = 4 × 6 and 24 = 6 × 4, so 24k = 4 × (6k) = 6 × (4k). Converse: if n is divisible by both 4 and 6, then n is divisible by 24. Counterexample: n = 12. It is divisible by 4 and by 6 but not by 24. The reason is that 4 and 6 share the factor 2; a number divisible by both need only be divisible by their lcm, 12.

A practical lesson. The last row explains why you cannot test divisibility by 8 by checking 2 and 4: the number 12 passes both tests but is not a multiple of 8. To test 8 you look at the last three digits.

Language. When a proposition is true and its converse false, the conclusion is a necessary consequence of the hypothesis, but the conclusion alone is not enough to bring back the hypothesis.

Sometimes a proposition and its converse are both true. Then X and Y always come together: whenever one holds, so does the other. We say ‘X if and only if Y’ and write X ⇔ Y.

A number fact. Let n be a positive whole number. Proposition P: if n is a perfect square, then n has an odd number of factors. Converse Q: if n has an odd number of factors, then n is a perfect square. Test a few cases: 12 has factors 1, 2, 3, 4, 6, 12 (six of them) and is not a square; 36 has factors 1, 2, 3, 4, 6, 9, 12, 18, 36 (nine) and is a square. Both statements look true. But examples are not proofs, so we need an argument.

Factor partners. Each factor f of n has a partner n ÷ f, and the two multiply to n. For 36 the pairs are (1, 36), (2, 18), (3, 12), (4, 9) and (6, 6). Factors therefore come in pairs, except when a factor is its own partner, f × f = n.

Proving Q. Suppose n has an odd number of factors. If every factor had a different partner, the factors would split into pairs and their number would be even. So some factor f is its own partner: f × f = n, and n is a perfect square. Notice what this argument does: it starts from ‘odd number of factors’ and ends at ‘perfect square’. It proves Q, not P.

Proving P. Now start from n = f × f. Then f is a factor that is its own partner. Can there be a second such factor g? Then g × g = n = f × f, and since both are positive, g = f. So exactly one factor is unpaired; all the others form pairs. The total is (an even number) + 1, which is odd. So P is true as well.

Conclusion. A positive whole number is a perfect square if and only if it has an odd number of factors. The two halves needed two different arguments. This is the usual pattern: to prove ‘if and only if’, you prove the proposition and then, separately, its converse.

Careful with squares. The third row is a classic trap. From x² = 9 you may not conclude x = 3, because x = −3 also works. In solving equations, a step that is only one-way can give you a wrong answer or lose an answer.

Geometry is full of propositions whose converses are also true, and proving those converses is one of the best uses of congruence. Here are two important cases.

1. The converse of the isosceles triangle theorem. You know: if AB = AC in ΔABC, then ∠B = ∠C. The converse says: if ∠B = ∠C, then AB = AC.

Proof. Draw the perpendicular AD from A to the side BC. Compare ΔABD and ΔACD.

• ∠ABD = ∠ACD (given).
• ∠ADB = ∠ADC = 90° (AD is perpendicular to BC).
• AD = AD (common side).

Two angles and a side that is not between them are equal, so ΔABD ≅ ΔACD by the AAS rule. Corresponding parts are equal, so AB = AC. The converse is true, and we can now say: two sides of a triangle are equal if and only if the angles opposite them are equal.

2. The converse of the Baudhāyana–Pythagoras theorem. The theorem says: in a triangle with sides a, b, c, if the angle between a and b is a right angle, then a² + b² = c². The converse says: if a² + b² = c², then the angle between the sides a and b is a right angle.

Proof. Let ΔABC have BC = a, CA = b, AB = c and a² + b² = c². Construct a second triangle XYZ with a right angle at Z, ZY = a and ZX = b. By the theorem, applied to ΔXYZ, XY² = a² + b². But a² + b² = c², so XY² = c² and XY = c. Now the two triangles have all three sides equal: BC = YZ, CA = ZX and AB = XY. By SSS, ΔABC ≅ ΔXYZ, so ∠C = ∠Z = 90°. The converse is true.

Using it. Is a triangle with sides 20, 21, 29 right-angled? 20² + 21² = 400 + 441 = 841 = 29². Yes, the right angle is opposite the side 29. With sides 6, 7, 9: 36 + 49 = 85, and 9² = 81, which is different, so the triangle has no right angle. This is exactly the knotted-rope test of ancient builders, now with a proof.

3. Parallel lines. If two lines are parallel, the corresponding angles made by a transversal are equal; and conversely, if the corresponding angles are equal, the lines are parallel. Both are true and are used together all the time.

The method. In each proof we built or drew something extra (a perpendicular, a second triangle) so that a congruence rule could be used. Copying the original proof backwards rarely works; a converse usually needs a fresh idea.

We have seen a true proposition with a false converse, and pairs where both are true. The remaining possibilities also happen. A false proposition can have a true converse, and it is even possible for both to be false.

Both false. Proposition: if n is a prime number, then n is odd. False, because 2 is prime and even. Converse: if n is odd, then n is prime. False, because 9 is odd and 9 = 3 × 3. Another pair: if a whole number is greater than 10, then it is even (false: 11); converse: if a whole number is even, then it is greater than 10 (false: 4). To show that both are false you need two counterexamples, one for each direction.

False proposition, true converse. ‘If a number is even, then it is divisible by 4’ is false (6); its converse ‘if a number is divisible by 4, then it is even’ is true, since 4k = 2 × (2k).

All four cases.

Divisibility shortcuts as propositions. Shortcuts are only useful if they work both ways. Look at the test for 3:

• If a number is divisible by 3, then the sum of its digits is a multiple of 3.
• If the sum of the digits of a number is a multiple of 3, then the number is divisible by 3.

Both are true, so: a number is divisible by 3 if and only if its digit sum is a multiple of 3. The reason is that 10, 100, 1000, … each leave remainder 1 when divided by 3. For example, 4572 = 4 × 1000 + 5 × 100 + 7 × 10 + 2, and this differs from 4 + 5 + 7 + 2 = 18 by a multiple of 3 (4 × 999 + 5 × 99 + 7 × 9). Since 18 is a multiple of 3, so is 4572; and 4572 ÷ 3 = 1524.

A shortcut that fails. ‘If n is divisible by 2 and by 4, then n is divisible by 8’ is false (n = 12), so it is not a valid test for 8. On the other hand, ‘if n is divisible by 3 and by 4, then n is divisible by 12’ is true, because 3 and 4 have no common factor other than 1.

Another number fact. Let n be a positive whole number. If n and n + 3 have no common factor other than 1, then n is not a multiple of 3. This is true: if n were a multiple of 3, so would n + 3 be, and 3 would be a common factor. Its converse, ‘if n is not a multiple of 3, then n and n + 3 have no common factor other than 1’, is also true: any common factor of n and n + 3 divides their difference 3, so it is 1 or 3, and 3 does not divide n.

Lesson. Decide each direction on its own; never let the truth of one direction decide the other.

A carpenter wants to build a rectangular window frame using two thin sticks as the diagonals. Should the sticks be equal? Does the way they are crossed matter? Questions like these turn on the difference between a proposition and its converse.

Two kinds of condition. In ‘if X, then Y’:

• Y is a necessary condition for X: whenever X holds, Y must hold. Without Y, X is impossible.
• X is a sufficient condition for Y: X is enough to guarantee Y.

If the converse is also true, each is both necessary and sufficient for the other; this is the meaning of ‘if and only if’.

Example: equal diagonals. Suppose a type of quadrilateral, call it type Q, has this property: if a quadrilateral is of type Q, then its diagonals are equal. Then equal diagonals are necessary for type Q. To build a Q from two sticks, the sticks must be equal. But this property says nothing about how to cross them; equal sticks may or may not give a Q, depending on how they are placed. Rectangles are like this: their diagonals are equal, but two equal sticks crossed at their ends, or crossed off-centre, need not give a rectangle. To get a rectangle you also need the sticks to cross at their midpoints. (A quadrilateral whose diagonals are equal and bisect each other is a rectangle.)

Now suppose instead: if a quadrilateral has equal diagonals, then it is of type Q. Then equal diagonals are sufficient. Any two equal sticks, crossed in any way that gives a quadrilateral, will produce a Q. The sticks still need to be equal for this guarantee to apply, but the crossing no longer matters.

Everyday language. ‘Passing the exam is necessary for admission’ means: if admitted, then passed. It does not mean that passing guarantees admission. ‘Scoring 90% is sufficient for a scholarship’ means: if 90%, then a scholarship. It does not mean scholarship holders all scored 90%.

Why mathematicians care. A necessary condition helps you rule things out quickly: a quadrilateral whose diagonals are unequal cannot be a rectangle. A sufficient condition helps you construct or prove things: to show a triangle is right-angled, it is sufficient to check a² + b² = c². Knowing which direction you have is the whole skill of this chapter.