1. Identify real-life examples in which a proposition is true but not the converse. Give nice counterexamples!
2. Give more examples from geometry as well as number theory in which a proposition is true but not its converse. Give appropriate counterexamples.
- 1. Identify real-life examples in which a proposition is true but not the converse. Give nice counterexamples!
- 2. Give more examples from geometry as well as number theory in which a proposition is true but not its converse. Give appropriate counterexamples.
Step-by-step solution
Idea: A proposition ‘if X then Y’ is true but its converse false when everything with property X has property Y, but Y is a bigger group. Any member of the bigger group outside X is a counterexample to the converse.
1. Identify real-life examples in which a proposition is true but not the converse. Give nice counterexamples!
- ‘If a person lives in Manama, the capital of Bahrain, then the person lives in Bahrain.’ True: Manama is in Bahrain. Converse: ‘If a person lives in Bahrain, then the person lives in Manama, the capital of Bahrain.’ Counterexample: someone who lives in Muharraq.½ mark
- ‘If a vehicle is a car, then it has wheels.’ True. Converse: ‘If a vehicle has wheels, then it is a car.’ Counterexample: a bicycle has wheels but is not a car.½ mark
- ‘If a person is 15 years old, then the person is a teenager.’ True. Converse: ‘If a person is a teenager, then the person is 15 years old.’ Counterexample: a 17-year-old.½ mark
- In each case the ‘then’ group (people in Bahrain, things with wheels, teenagers) is bigger than the ‘if’ group, so the converse fails for anyone in the bigger group but outside the smaller one.½ mark
2. Give more examples from geometry as well as number theory in which a proposition is true but not its converse. Give appropriate counterexamples.
- Geometry: ‘If a quadrilateral is a rectangle, then its diagonals are equal.’ True. Converse: ‘If a quadrilateral has equal diagonals, then it is a rectangle.’ Counterexample: an isosceles trapezium has equal diagonals but its angles are not all 90°.½ mark
- Geometry: ‘If two angles are vertically opposite, then they are equal.’ True. Converse: ‘If two angles are equal, then they are vertically opposite.’ Counterexample: the two base angles of an isosceles triangle are equal but are not vertically opposite.½ mark
- Number theory: ‘If a and b are both even, then a + b is even.’ True: 2x + 2y = 2(x + y). Converse: ‘If a + b is even, then a and b are both even.’ Counterexample: 3 + 5 = 8.½ mark
- Number theory: ‘If n is a prime greater than 2, then n is odd.’ True. Converse: ‘If n is odd, then n is a prime greater than 2.’ Counterexample: 9 is odd, but 9 = 3 × 3 is not prime.½ mark
Answer to write in the exam
1.
P: If a person lives in Manama, the capital of Bahrain, then the person lives in Bahrain. (True)
Converse: If a person lives in Bahrain, then the person lives in Manama, the capital of Bahrain. False: a person living in Muharraq.
P: If a vehicle is a car, then it has wheels. (True)
Converse: If a vehicle has wheels, then it is a car. False: a bicycle.
P: If a person is 15 years old, then the person is a teenager. (True)
Converse: If a person is a teenager, then the person is 15. False: a 17-year-old.
2.
P: If a quadrilateral is a rectangle, then its diagonals are equal. (True)
Converse false: an isosceles trapezium has equal diagonals but is not a rectangle.
P: If two angles are vertically opposite, then they are equal. (True)
Converse false: base angles of an isosceles triangle are equal but not vertically opposite.
P: If a and b are even, then a + b is even: 2x + 2y = 2(x + y). (True)
Converse false: 3 + 5 = 8 is even, but 3 and 5 are odd.
P: If n is a prime greater than 2, then n is odd. (True)
Converse (if n is odd, then n is a prime greater than 2) false: 9 is odd but 9 = 3 × 3 is not prime.
Common mistakes that cost marks
- Choosing a proposition whose converse is also true, such as ‘If a triangle has three equal sides, then it has three equal angles’.
- Giving a counterexample to the proposition instead of to the converse.
- Starting from a proposition that is not always true, such as ‘If it is cloudy, then it will rain’. The proposition itself must be true.
How this can come in the exam
Which proposition is true but has a false converse?
- If a triangle is equilateral, then each of its angles is 60°.
- If a number is a multiple of 8, then it is even.
- If x + 3 = 7, then x = 4.
- If a quadrilateral is a parallelogram, then its opposite sides are equal.
Show answer
(B) If a number is a multiple of 8, then it is even.
Every multiple of 8 is even (8k = 2 × 4k), but 6 is even and not a multiple of 8. The other three have true converses.
Write a true proposition about rhombuses whose converse is false, and give a counterexample to the converse.
Show answer
‘If a quadrilateral is a rhombus, then its diagonals are perpendicular’ — true (1 mark). Converse: ‘If the diagonals of a quadrilateral are perpendicular, then it is a rhombus’ — false: a kite with sides 3, 3, 5, 5 cm has perpendicular diagonals but is not a rhombus (1 mark).Try one yourself
Write the converse of ‘If a number is a multiple of 100, then it ends in 0’, and give a counterexample to it.
Show answer
Converse: ‘If a number ends in 0, then it is a multiple of 100.’ False: 30 ends in 0 but is not a multiple of 100.
More questions like this
- If two lines are parallel, then the corresponding angles formed by a transversal are equal.
- If a quadrilateral is a square, then all its angles are equal.
- Given any ∆ABC, let us bisect the angles at B and C. The bisectors meet at the incentre I of the triangle.
Now extend the bisectors beyond I till they meet the opposite sides at E and F respectively, as shown.
Proposition: If AB = AC, then IE = IF. - If x = y, then a + x = a + y, where x, y and a are any three numbers.
This proposition and its converse are routinely used while solving equations. - If a and b are perfect squares, then ab is a perfect square.