Propositions and if-then statements
A teacher says "If you finish the work, you may go out." One part gives a condition, the other a result.
A proposition is a sentence that is either true or false. An if-then statement "If P, then Q" has a hypothesis P (given) and a conclusion Q (to be shown). It is written P ⇒ Q.
Formula: If P, then Q (P ⇒ Q)
In the figure: P = "6 divides n" is the hypothesis, Q = "2 divides n" is the conclusion.
Rule 1: Find P and Q — P follows "if", Q follows "then". Rewrite hidden statements in this form.
- Easy: If it rains, the road is wet: P = it rains, Q = road is wet.
- Medium: "A square is a rectangle" ⇒ if a shape is a square, then it is a rectangle.
- Exam-type: "x = 3 gives x² = 9" ⇒ if x = 3 then x² = 9.
Rule 2: True or false? — A statement is true only if it is true every time the hypothesis holds.
- Easy: If a number is a multiple of 4, it is even: true.
- Medium: If a number is even, it is a multiple of 4: false (6).
- Exam-type: If a triangle is equilateral, it is isosceles: true.
One failing case is enough to make an if-then statement false.
Mixing up the hypothesis and the conclusion.
"IF is given, THEN is to prove."
Practice questions
- 1In the proposition "If a triangle is right-angled, then the square of its longest side equals the sum of the squares of the other two sides.", what is the hypothesis (the part after 'if')?
- 2In the proposition "If a number is divisible by 4, then it is even.", what is the hypothesis (the part after 'if')?
- 3In the proposition "If a number is divisible by 9, then it is divisible by 3.", what is the hypothesis (the part after 'if')?
- 4In the proposition "If a number ends in 0, then it is divisible by 10.", what is the conclusion (the part after 'then')?
- 5In the proposition "If a number is a multiple of 8, then it is a multiple of 4.", what is the conclusion (the part after 'then')?
- 6In the proposition "If a line from the centre of a circle is perpendicular to a chord, then it bisects the chord.", what is the conclusion (the part after 'then')?
- 7In the proposition "If a number is an even number, then its square is even.", what is the hypothesis (the part after 'if')?
- 8In the proposition "If a number is a multiple of 15, then it is a multiple of 3 and also a multiple of 5.", what is the hypothesis (the part after 'if')?
Show answers (try them yourself first)
- 1a triangle is right-angled
- 2a number is divisible by 4
- 3a number is divisible by 9
- 4it is divisible by 10
- 5it is a multiple of 4
- 6it bisects the chord
- 7a number is an even number
- 8a number is a multiple of 15
Forming the converse
Looking at a story backwards: "If it rains, the ground is wet" turns into "If the ground is wet, it rained". That is the converse.
The converse of "If P, then Q" is "If Q, then P": swap the hypothesis and the conclusion. A statement and its converse are different statements.
Formula: P ⇒ Q has converse Q ⇒ P
In the figure: The converse reads the same arrow from the other side: Q ⇒ P.
Rule 1: Write the converse — Swap "if" and "then" parts without changing their wording.
- Easy: If x = 2, then x + 3 = 5. Converse: if x + 3 = 5, then x = 2.
- Medium: If a number ends in 0, it is divisible by 10. Converse: if divisible by 10, it ends in 0.
- Exam-type: If it is a square, all sides are equal. Converse: if all sides are equal, it is a square.
Rule 2: Hidden if-then forms — Rewrite "All A are B" as "If it is A, then it is B".
- Easy: All squares are rectangles ⇒ If square, then rectangle.
- Medium: Converse: If rectangle, then square.
- Exam-type: "Only if" reverses the roles: "P only if Q" means P ⇒ Q.
The converse of the converse is the original statement.
Writing the converse by just negating both parts.
"Flip the arrow."
Practice questions
- 1What is the converse of "If x² = 16, then x = 4."?
- 2In the converse of "If x² = 16, then x = 4.", what is the hypothesis?
- 3In the converse of "If a number is a multiple of 12, then it is a multiple of 6.", what is the hypothesis?
- 4In the converse of "If x = 3, then x² = 9.", what is the hypothesis?
- 5In the converse of "If two triangles are congruent, then they have equal areas.", what is the hypothesis?
- 6In the converse of "If a number is prime, then it is odd.", what is the hypothesis?
- 7The converse of 'if X then Y' is
- 8The converse of 'X implies Y' is
Show answers (try them yourself first)
- 1If x = 4, then x² = 16.
- 2x = 4
- 3it is a multiple of 6
- 4x² = 9
- 5they have equal areas
- 6it is odd
- 7if Y then X
- 8Y implies X
Is the converse true?
"If it is a dog, it has four legs" is true, but "If it has four legs, it is a dog" is not: cats and cows also have four legs.
A true statement may have a false converse or a true converse. Each must be checked on its own. When both are true we write P ⇔ Q ("P if and only if Q").
Formula: P ⇒ Q true does not make Q ⇒ P true
In the figure: The original is true, but its converse fails for cats, so the converse is false.
Rule 1: Test both directions — Look for one example where the converse fails.
- Easy: If x = 3 then x² = 9 (true). Converse false: x = −3.
- Medium: Vertical angles are equal (true); equal angles need not be vertical (false).
- Exam-type: A triangle with three equal sides has three equal angles, and the converse also holds: true both ways.
Rule 2: If and only if — Use ⇔ only when both directions are true.
- Easy: A number is even if and only if it is divisible by 2.
- Medium: A quadrilateral is a rectangle iff it is a parallelogram with a right angle.
- Exam-type: x + 2 = 5 iff x = 3.
To prove "iff" you must prove both P ⇒ Q and Q ⇒ P.
Assuming that if a statement is true, its converse is true too.
"Check both ways before saying iff."
Practice questions
- 1If a proposition is true, its converse is
- 2A single example that satisfies the hypothesis but violates the conclusion is called a
- 3The proposition "If a number is a multiple of 8, then it is a multiple of 4." is true. Is its converse "If it is a multiple of 4, then a number is a multiple of 8." also true?
- 4The proposition "If a child is in Class 9, then the child is older than 10 years." is true. Is its converse "If the child is older than 10 years, then a child is in Class 9." also true?
- 5The proposition "If a number is divisible by 100, then its last two digits are 00." is true. Is its converse "If its last two digits are 00, then a number is divisible by 100." also true?
- 6The proposition "If x = 3, then x² = 9." is true. Is its converse "If x² = 9, then x = 3." also true?
- 7Look at "If a number is a multiple of 12, then it is a multiple of 6." and its converse "If it is a multiple of 6, then a number is a multiple of 12." Which description is correct?
- 8Look at "If x² = 16, then x = 4." and its converse "If x = 4, then x² = 16." Which description is correct?
Show answers (try them yourself first)
- 1not necessarily true
- 2counterexample
- 3No
- 4No
- 5Yes
- 6No
- 7the proposition is true but its converse is false
- 8the proposition is false but its converse is true
Counterexamples
A shopkeeper claims "all mangoes in this box are sweet". One sour mango proves the claim wrong.
A counterexample is one case where the hypothesis holds but the conclusion fails. One counterexample disproves a statement. Many examples that work do not prove it.
Formula: One counterexample ⇒ statement is false
In the figure: The prime 2 is even, so it breaks the claim "all primes are odd".
Rule 1: Find a counterexample — Try small and special numbers: 0, 1, 2, negatives, fractions.
- Easy: "n² > n for all n": n = 1 gives 1 = 1.
- Medium: "Sum of two primes is even": 2 + 3 = 5.
- Exam-type: "(a + b)² = a² + b²": a = b = 1 gives 4 ≠ 2.
Rule 2: Examples are not proofs — To prove a statement you need an argument that works for every case.
- Easy: n² + n + 41 is prime for n = 0 to 39, but n = 40 gives 41² (not prime).
- Medium: Five true examples do not prove a rule.
- Exam-type: A single failing case is a valid disproof.
A good counterexample is specific and easy to check.
Giving a case where the statement holds and calling it a counterexample.
"One bad apple spoils the claim."
Practice questions
- 1Claim: "A number divisible by both 4 and 6 is divisible by 24." Is 12 a counterexample?
- 2What is the smallest natural number that disproves the claim "Every multiple of 5 is a multiple of 15"?
- 3What is the smallest natural number that disproves the claim "Every multiple of 3 is a multiple of 9"?
- 4Claim: "If n² is divisible by 25, then n is divisible by 25." Which value of n is a counterexample?
- 5Claim: "If n² is divisible by 36, then n is divisible by 36." Which value of n is a counterexample?
- 6Claim: "If a number is a multiple of both 10 and 25, then it is a multiple of 250." Which number is a counterexample?
- 7Claim: "If a number is a multiple of both 8 and 12, then it is a multiple of 96." Which number is a counterexample?
- 8Claim: "(a + b)² = a² + b² for all numbers." Which pair (a, b) is a counterexample?
Show answers (try them yourself first)
- 1Yes, 12 is a counterexample
- 25
- 33
- 45
- 56
- 650
- 724
- 8a = 3, b = 5
Converses of geometry theorems
Equal sides give equal angles in a triangle. Does equal angles give equal sides? Geometry asks this about every theorem.
Some theorem converses are true: isosceles triangle theorem (equal sides ⇒ equal opposite angles) and its converse; Pythagoras and its converse. Others are false: "vertical angles are equal" has a false converse.
Formula: a² + b² = c² ⇔ right angle at the vertex opposite c
In the figure: Sides 3, 4, 5 satisfy 3² + 4² = 5², so by the converse the triangle is right-angled.
Rule 1: True converses — Use them as tests: from sides, decide angles, and the reverse.
- Easy: Two equal angles in a triangle ⇒ the opposite sides are equal.
- Medium: 5, 12, 13: 25 + 144 = 169, right-angled.
- Exam-type: Alternate angles equal ⇒ lines parallel.
Rule 2: False converses — Find one case where the converse breaks.
- Easy: Congruent triangles have equal areas; equal areas need not mean congruent.
- Medium: A square has equal diagonals; equal diagonals do not give a square (rectangle).
- Exam-type: 7, 8, 12: 49 + 64 = 113 ≠ 144, not right-angled.
Many key geometry tests (SSS, ASA, parallel-line tests) are converses of earlier theorems.
Using the converse of a theorem without proving it holds.
"Check the arrow in both directions."
Practice questions
- 1In quadrilateral PQRS, PQ = RS = 11 cm and QR = SP = 9 cm. What can we conclude?
- 2In quadrilateral PQRS, PQ = RS = 12 cm and QR = SP = 4 cm. What can we conclude?
- 3In triangle ABC, ∠B = ∠C = 68°. Using the converse of the isosceles triangle theorem, which two sides are equal?
- 4In triangle ABC, ∠B = ∠C = 34°. Using the converse of the isosceles triangle theorem, which two sides are equal?
- 5In triangle ABC, ∠B = ∠C = 59°. Using the converse of the isosceles triangle theorem, which two sides are equal?
- 6A triangle has sides 6 cm, 8 cm and 10 cm. Using the converse of the Baudhāyana–Pythagoras theorem, is it right-angled?
- 7In quadrilateral ABCD, ∠A = 120° and ∠C = 60°. By the converse of the cyclic quadrilateral theorem, what can we say?
- 8In quadrilateral ABCD, ∠A = 75° and ∠C = 105°. By the converse of the cyclic quadrilateral theorem, what can we say?
Show answers (try them yourself first)
- 1PQRS is a parallelogram
- 2PQRS is a parallelogram
- 3AB and AC
- 4AB and AC
- 5AB and AC
- 6Yes, it is right-angled
- 7ABCD is cyclic
- 8ABCD is cyclic