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Propositions and their Converses - Different Ways of Expressing If X then Y

Grade 9CBSE

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

🔑Concepts

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A proposition (or statement) is a declarative sentence that is either true or false, but not both.

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A conditional statement is a compound statement of the form 'If pp, then qq', where pp and qq are individual statements.

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The statement pp is called the hypothesis or antecedent, and qq is called the conclusion or consequent.

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The converse of the statement 'If pp, then qq' is the statement '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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When a statement 'If pp, then qq' and its converse 'If qq, then pp' are both true, we say 'pp if and only if qq' (denoted as p  ⟺  qp \iff q).

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There are several ways to express the conditional statement p  ⟹  qp \implies q in English:

  1. If pp, then qq.
  2. pp implies qq.
  3. pp is sufficient for qq.
  4. qq is necessary for pp.
  5. pp only if qq.
  6. qq if pp.

📐Formulae

p  ⟹  q (Conditional: If p, then q)p \implies q \text{ (Conditional: If } p, \text{ then } q)

q  ⟹  p (Converse: If q, then p)q \implies p \text{ (Converse: If } q, \text{ then } p)

p  ⟺  q (Biconditional: p if and only if q)p \iff q \text{ (Biconditional: } p \text{ if and only if } q)

💡Examples

Problem 1:

Write the converse of the following statement and determine if the converse is true: 'If a number is divisible by 1010, then it is divisible by 55.'

Solution:

The original statement is of the form 'If pp, then qq'.

  • pp: A number is divisible by 1010.
  • qq: A number is divisible by 55.

The converse is 'If qq, then pp': 'If a number is divisible by 55, then it is divisible by 1010.'

Truth Value of Converse: The converse is false. Counter-example: The number 1515 is divisible by 55, but it is not divisible by 1010.

Explanation:

To form the converse, we swap the hypothesis and the conclusion. While the original statement is true (multiples of 1010 are always multiples of 55), the reverse is not always true.

Problem 2:

Rewrite the statement 'A quadrilateral is a square only if it is a rectangle' in the form 'If pp, then qq'.

Solution:

The phrase 'pp only if qq' is logically equivalent to 'If pp, then qq'.

  • pp: A quadrilateral is a square.
  • qq: It is a rectangle.

Result: 'If a quadrilateral is a square, then it is a rectangle.'

Explanation:

In logic, 'pp only if qq' means that pp cannot happen without qq happening. Thus, if pp is true, qq must be true.

Problem 3:

Given the statement: 'If the sum of digits of a number is divisible by 33, then the number is divisible by 33'. Identify the hypothesis, the conclusion, and write the converse.

Solution:

Hypothesis (pp): The sum of digits of a number is divisible by 33. Conclusion (qq): The number is divisible by 33.

Converse: 'If a number is divisible by 33, then the sum of its digits is divisible by 33.'

Explanation:

In this specific case (divisibility rule of 33), both the statement and its converse are true, meaning the condition is 'if and only if'.