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Propositions and their Converses - Converses in Number Theory

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 logical statement of the form 'If PP, then QQ', where PP is the hypothesis and QQ is the conclusion. This is denoted as P  ⟹  QP \implies Q.

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The converse of a proposition 'If PP, then QQ' is formed by swapping the hypothesis and the conclusion: 'If QQ, then PP'. This is denoted as Q  ⟹  PQ \implies P.

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In Number Theory, if a statement is true, its converse is not necessarily true. For example, 'If a number is divisible by 44, then it is even' is true, but its converse 'If a number is even, then it is divisible by 44' is false (counter-example: 22 or 66).

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To prove a converse is false, we must provide a counter-example. A counter-example is a case where the hypothesis of the converse is true, but the conclusion is false.

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If both a proposition and its converse are true, the conditions PP and QQ are said to be equivalent, often written as 'PP if and only if QQ' (P  ⟺  QP \iff Q).

📐Formulae

P  ⟹  Q (Proposition)P \implies Q \text{ (Proposition)}

Q  ⟹  P (Converse)Q \implies P \text{ (Converse)}

P  ⟺  Q (Biconditional/Equivalence)P \iff Q \text{ (Biconditional/Equivalence)}

💡Examples

Problem 1:

Given the proposition: 'If a natural number nn is divisible by 1010, then it ends with the digit 00'. Write its converse and determine if the converse is true.

Solution:

  1. Identify PP and QQ: PP: nn is divisible by 1010 QQ: nn ends with the digit 00

  2. Form the converse (Q  ⟹  PQ \implies P): 'If a natural number nn ends with the digit 00, then it is divisible by 1010'.

  3. Check truth value: Every number ending in 00 (e.g., 10,20,15010, 20, 150) is a multiple of 1010. Therefore, the converse is True.

Explanation:

In this case, the property of being divisible by 1010 is equivalent to ending in the digit 00 in the decimal system.

Problem 2:

State the converse of the proposition: 'If xx is a prime number, then xx is an odd number'. Provide a counter-example if the converse is false.

Solution:

  1. Proposition: 'If xx is prime, then xx is odd'. (Note: The original proposition is actually false because x=2x=2 is prime but not odd, but we are asked for the converse of the given statement).

  2. Converse: 'If xx is an odd number, then xx is a prime number'.

  3. Truth Value: False.

  4. Counter-example: Let x=9x = 9. x=9x = 9 is an odd number (hypothesis of converse is true). But x=9x = 9 is not a prime number because its factors are 1,3,91, 3, 9 (conclusion of converse is false).

Explanation:

A single counter-example like 99, 1515, or 2121 is sufficient to show that the converse 'All odd numbers are prime' is false.

Problem 3:

Consider the proposition: 'If nn is an even natural number, then n2n^2 is divisible by 44'. State the converse and verify it.

Solution:

  1. Converse: 'If n2n^2 is divisible by 44, then nn is an even natural number'.

  2. Verification: Let n2n^2 be divisible by 44. This means n2=4kn^2 = 4k for some integer kk. Taking the square root, n=4k=2kn = \sqrt{4k} = 2\sqrt{k}. Since nn is a natural number, kk must be a perfect square, say k=m2k = m^2. Then n=2mn = 2m, which is the definition of an even number.

  3. Conclusion: The converse is True.

Explanation:

Since n2n^2 being a multiple of 44 implies nn must contain the factor 22, nn must be even.