Class 9 Maths Chapter 9: Propositions and their Converses — Notes, Examples & Practice

A proposition is a sentence that is definitely true or false, not both. '7 is a prime number' is a proposition (true). '12 is odd' is a proposition (false). 'Close the door' and 'Is it raining?' are not propositions.

Many mathematical propositions have the form 'If X, then Y', written $X \Rightarrow Y$ and read 'X implies Y'. X is the hypothesis (what we assume) and Y the conclusion. It claims that whenever X is true, Y is true as well.

Statements can be reworded into this form: 'Every multiple of 6 is a multiple of 3' means 'If a number is a multiple of 6, then it is a multiple of 3'.

The converse of 'If X, then Y' is 'If Y, then X' — the hypothesis and conclusion swap places.

A true statement can have a false converse. 'If it is raining, the road is wet' is true, but 'If the road is wet, it is raining' is false — someone may have washed it. Similarly 'If $n$ is a multiple of 6, then it is a multiple of 3' is true, but its converse fails at $n = 9$.

So you must judge a converse separately. Never assume it is true because the original is.

A claim about all cases is disproved by one case where it fails — a counterexample. To disprove 'Every prime is odd', the single example 2 is enough.

But a claim about all cases is not proved by checking a few examples, however many. $n^2 + n + 41$ gives primes for $n = 0, 1, \dots, 39$, yet at $n = 40$ it equals $40^2 + 40 + 41 = 41^2$, not prime.

History has a famous case: Fermat claimed every number $2^{2^n} + 1$ is prime. It is for $n = 0$ to 4 (3, 5, 17, 257, 65537), but Euler showed $2^{32} + 1 = 641 \times 6700417$.

Sometimes a statement and its converse are both true. Then X and Y are equivalent, written $X \Leftrightarrow Y$ and read 'X if and only if Y'.

Example: 'A positive integer is a perfect square if and only if it has an odd number of factors.' Why: factors come in pairs $(d, \dfrac{n}{d})$. The pairs always contribute an even count — except when $d = \dfrac{n}{d}$, i.e. $d^2 = n$, which happens only for perfect squares. So 36 has factors 1, 2, 3, 4, 6, 9, 12, 18, 36 (nine of them, because 6 pairs with itself).

Not every 'nice' converse works: $x^3 = y^3 \Rightarrow x = y$ is true for real numbers, but $x^2 = y^2 \Rightarrow x = y$ is false ($x = 3$, $y = -3$).

Baudhāyana–Pythagoras theorem: if a triangle has a right angle, then the square on the hypotenuse equals the sum of the squares on the other two sides ($a^2 + b^2 = c^2$). Baudhāyana's Śulba-sūtra stated this for rope constructions of altars.

Its converse is also true: if the sides of a triangle satisfy $a^2 + b^2 = c^2$, then the angle opposite $c$ is $90^\circ$. Proof idea: build a right triangle with legs $a$ and $b$; its hypotenuse is $\sqrt{a^2 + b^2} = c$. The two triangles have the same three sides, so they are congruent by SSS, and the original angle must be $90^\circ$ too.

Isosceles triangle theorem: if two sides are equal, the angles opposite them are equal. Converse (true): if two angles are equal, the sides opposite them are equal. Proof idea: draw the bisector (or the perpendicular) from the third vertex and use AAS congruence.