Propositions and their Converses

Chapter at a glance

Chapter 9 opens Ganita Manjari Part II with the language of mathematical reasoning: propositions, their converses and counterexamples. Chapter 12 (Quadrilaterals) depends on these ideas, so learn them well.

What is a proposition?

Example: “If two sides of a triangle are equal, then the angles opposite the equal sides are equal.”

Converse

The converse of the example is: “If two angles of a triangle are equal, then the sides opposite them are equal.” (Also true. Hint for the proof: draw the altitude from the third vertex.)

Ways of saying the same thing: “if X then Y” = “X implies Y” = “Y when X”. For example, “A number has an odd number of factors when it is a perfect square” means the same as “If a number is a perfect square, then it has an odd number of factors.”

Counterexamples

• Rain: “If it rains, the road is wet” is true. Its converse “If the road is wet, it has rained” is false: a water tanker may have spilled water.
• Fermat numbers: Fermat claimed every number of the form 2²ⁿ + 1 is prime (3, 5, 17, 257, 65537, …). Euler showed 2³² + 1 = 641 × 6700417 is composite.

Worked examples from the chapter

Why perfect squares have an odd number of factors

Factors come in factor–partner pairs whose product is the number: for 12 they are (1, 12), (2, 6), (3, 4).

• Proving Q: odd number of factors ⇒ some pair repeats, (f, f) ⇒ n = f × f is a perfect square.
• Proving P: a number can have at most one repeating pair, so a square has exactly one factor counted once and every other factor paired, giving an odd total.

Lesson: an argument often proves only one direction. Check carefully which statement it really proves.

Proving the converse of Baudhāyana–Pythagoras

Suppose ∆ABC has sides a, b, c with a² + b² = c². Build a right triangle XYZ with legs a and b. By the theorem, XY² = a² + b² = c², so XY = c. Then ∆ABC ≅ ∆XYZ by SSS, so ∆ABC is right-angled too.

The possibilities

• P true, converse false (multiples of 6 and 3)
• Both true (perfect squares and an odd number of factors)
• P false, converse true (equal area and congruence)
• Both false: “If n is even, then n is a multiple of 3” (false: 4) and its converse “If n is a multiple of 3, then n is even” (false: 3).

Practice (Exercise Set 9.1)

For each, write the converse, decide whether each statement is true, and give a proof or a counterexample.

1. If two lines are parallel, the corresponding angles made by a transversal are equal. Both true.
2. If a quadrilateral is a square, all its angles are equal. P true; converse false (a non-square rectangle).
3. If x = y then a + x = a + y. Both true; used all the time when solving equations.
4. If a and b are perfect squares, then ab is a perfect square. P true; converse false (2 × 8 = 16).
5. If n is divisible by 24, it is divisible by both 4 and 6. P true; converse false (12).
6. If n is divisible by 60, it is divisible by both 5 and 12. Both true, since 5 and 12 share no common factor and 5 × 12 = 60.
7. If n is the square of a prime, it has exactly 3 factors. Both true.
8. Counterexamples to “all numbers of the form n² + n + 11 are prime”. n = 10 gives 121 = 11².
9. Counterexample to “If n is prime then 2ⁿ − 1 is prime”. n = 11: 2047 = 23 × 89.
10. Is checking divisibility by 2 and 4 enough to test divisibility by 8? No: 12 is divisible by 2 and 4 but not by 8.

Chapter summary

• A proposition is a statement that is true or false.
• The converse of “if P then Q” is “if Q then P”.
• A proposition can be true while its converse is false; both can be true; both can be false.
• Counterexamples show that a proposition is false.