MATHEMATICS • MATHEMATICAL REASONING
What Are Propositions and Their Converse?
A proposition is a statement that is either true or false. Its converse is formed by interchanging the condition and the conclusion of a conditional statement.
These ideas help us decide whether a mathematical statement is correct and whether its reverse is also correct.
1. What Is a Proposition?
A proposition is a declarative sentence that has exactly one truth value: true or false, but not both.
Think of a proposition as a sentence for which we can clearly decide whether it is correct or incorrect.
Examples of propositions
- “2 + 3 = 5” — True.
- “10 is an odd number” — False.
- “Every square has four sides” — True.
- “15 is divisible by 4” — False.
Sentences that are not propositions
- “Close the door.” — This is a command, not a statement that is true or false.
- “What is your name?” — This is a question.
- “This is a beautiful picture.” — Without a defined standard, “beautiful” is subjective.
- “x + 2 = 7” — If x has not been specified, this is an open sentence. It becomes a proposition once a value or suitable quantifier is given, for example, “There exists a real number x such that x + 2 = 7.”
2. What Is a Conditional Statement?
A conditional statement connects a condition with a conclusion. It is commonly written as “If p, then q”, where p is the hypothesis (condition) and q is the conclusion.
Conditional statement: If p, then q.
Symbolic form: p → q
For example:
If a number is divisible by 4, then it is even.
- Hypothesis (p): The number is divisible by 4.
- Conclusion (q): The number is even.
This statement is true for integers because every integer divisible by 4 is also divisible by 2.
3. What Is the Converse of a Proposition?
The converse of “If p, then q” is obtained by swapping p and q. It becomes “If q, then p.”
Original: If p, then q. (p → q)
Converse: If q, then p. (q → p)
For the example above:
- Original: If a number is divisible by 4, then it is even.
- Converse: If a number is even, then it is divisible by 4.
The original statement is true, but its converse is false. For example, 6 is even but is not divisible by 4. A single valid counterexample is enough to show that a universal statement is false.
4. More Examples with Truth Values
| Original statement | Converse | Original | Converse |
|---|---|---|---|
| If a number is divisible by 10, then it ends in 0. | If a number ends in 0, then it is divisible by 10. | True | True (for integers written in base 10) |
| If a shape is a square, then it has four equal sides. | If a shape has four equal sides, then it is a square. | True | False: a rhombus need not have four right angles. |
| If a number is divisible by 6, then it is divisible by 3. | If a number is divisible by 3, then it is divisible by 6. | True | False: 9 is divisible by 3 but not by 6. |
| If a figure is an equilateral triangle, then all its sides are equal. | If all sides of a triangle are equal, then it is equilateral. | True | True (by definition) |
| If a number is greater than 10, then it is greater than 5. | If a number is greater than 5, then it is greater than 10. | True | False: 7 is greater than 5 but not greater than 10. |
5. How to Write a Converse: Step by Step
- Identify the condition (hypothesis) after “if”.
- Identify the conclusion after “then”.
- Interchange the condition and the conclusion.
- Write the new statement and test whether it is true.
Example: If a triangle is equilateral, then it is isosceles.
Step 1 — Condition: The triangle is equilateral.
Step 2 — Conclusion: The triangle is isosceles.
Step 3 — Converse: If a triangle is isosceles, then it is equilateral.
Truth value: False. A triangle with side lengths 5 cm, 5 cm and 6 cm is isosceles but not equilateral.
6. Proposition, Converse, Inverse and Contrapositive
These terms are related but they mean different things. For a conditional statement “If p, then q”:
| Form | Statement | Symbolic form |
|---|---|---|
| Original | If p, then q. | p → q |
| Converse | If q, then p. | q → p |
| Inverse | If not p, then not q. | ¬p → ¬q |
| Contrapositive | If not q, then not p. | ¬q → ¬p |
Key fact: A conditional statement is logically equivalent to its contrapositive. The converse is logically equivalent to the inverse. The original and converse may have different truth values.
7. Everyday-Life Example
Suppose someone says:
“If a person is a mother, then that person is a parent.”
This is true. Its converse is:
“If a person is a parent, then that person is a mother.”
This is false because a parent may be a father. The example shows why reversing a true statement can produce a false one.
8. Practice Questions
- Is “17 is a prime number” a proposition? State its truth value.
- Explain why “Please sit down” is not a proposition.
- Write the converse: “If a number is divisible by 8, then it is even.”
- Is the converse in Question 3 true? Give a counterexample if it is false.
- Write the converse: “If a triangle is equilateral, then it is equiangular.” Is it true?
- Write the converse of “If a number is a multiple of 12, then it is a multiple of 3.”
Answers
- Yes. It is true.
- It is a command, not a statement with a truth value.
- If a number is even, then it is divisible by 8.
- No. 6 is even but not divisible by 8.
- If a triangle is equiangular, then it is equilateral. True for triangles in Euclidean geometry.
- If a number is a multiple of 3, then it is a multiple of 12. This is false; for example, 6 is a multiple of 3 but not of 12.
Frequently Asked Questions
Can a false statement be a proposition?
Yes. A proposition must have a definite truth value, but that value can be either true or false.
Is the converse always true when the original statement is true?
No. For example, “If a number is divisible by 4, then it is even” is true, but its converse is false.
What is a counterexample?
A counterexample is one specific example that proves a general statement false.
What is the difference between a converse and a contrapositive?
The converse swaps the hypothesis and conclusion. The contrapositive swaps them and negates both. A statement and its contrapositive always have the same truth value.