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Propositions and their Converses - Counterexamples

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A proposition or conditional statement is a statement of the form 'If PP, then QQ', often written as P  ⟹  QP \implies Q. Here, PP is the hypothesis and QQ is the conclusion.

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The converse of a statement 'If PP, then QQ' is 'If QQ, then PP'.

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It is important to note that if a statement is true, its converse is not necessarily true.

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A counterexample is a specific case or example that shows a general statement is false. To disprove a statement, finding a single counterexample is sufficient.

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In geometry, many theorems have converses that are also theorems (e.g., Isosceles Triangle Theorem), but this is not a universal rule.

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To show that the statement 'If QQ, then PP' is false, we must find an instance where QQ is satisfied (true) but PP is not satisfied (false).

📐Formulae

Original Statement: P  ⟹  Q\text{Original Statement: } P \implies Q

Converse Statement: Q  ⟹  P\text{Converse Statement: } Q \implies P

Logical Inequality: (P  ⟹  Q)≠(Q  ⟹  P) (in general)\text{Logical Inequality: } (P \implies Q) \neq (Q \implies P) \text{ (in general)}

💡Examples

Problem 1:

Consider the statement: 'If a number nn is prime, then nn is odd.' Write its converse and provide a counterexample to show why the original statement is false.

Solution:

Original Statement: If nn is prime   ⟹  n\implies n is odd. Converse Statement: If nn is odd   ⟹  n\implies n is prime.

Counterexample for the original statement: The number n=2n = 2 is a prime number, but it is not odd (it is even). Since we found one case where the hypothesis is true but the conclusion is false, the statement is false.

Explanation:

A counterexample must satisfy the 'if' part but fail the 'then' part.

Problem 2:

Given the statement: 'If a quadrilateral is a square, then it is a rectangle.' Write the converse and determine if the converse is true or false. If false, provide a counterexample.

Solution:

Original Statement: If a quadrilateral is a square, then it is a rectangle. (True) Converse Statement: If a quadrilateral is a rectangle, then it is a square.

Status: The converse is False. Counterexample: Consider a rectangle with length l=5 cml = 5 \text{ cm} and breadth b=3 cmb = 3 \text{ cm}. In this case, the quadrilateral is a rectangle (opposite sides are equal, all angles are 90∘90^\circ), but it is not a square because all four sides are not equal.

Explanation:

Even though every square is a rectangle, not every rectangle is a square. The counterexample highlights the specific property (equal sides) that is missing.

Problem 3:

Write the converse of the statement: 'If x2=25x^2 = 25, then x=5x = 5'. Is the original statement true? Provide a counterexample if it is false.

Solution:

Original Statement: If x2=25x^2 = 25, then x=5x = 5. Converse Statement: If x=5x = 5, then x2=25x^2 = 25.

Truth Value of Original Statement: False. Counterexample: Let x=−5x = -5. Then x2=(−5)2=25x^2 = (-5)^2 = 25. Here, the hypothesis x2=25x^2 = 25 is true, but the conclusion x=5x = 5 is false. Therefore, the original statement is false.

Explanation:

In algebra, square roots have both positive and negative values. Finding the negative value serves as a counterexample to the claim that xx must be positive 55.