Mathematical Reasoning Ex 14.3 Class 11 NCERT

Understanding Quantifiers: 'For All' and 'There Exists'

In mathematics, we often deal with statements that are not about a single object, but about a whole collection of objects. An 'open sentence' like 'x is a prime number' is not a statement because its truth depends on the value of x. Quantifiers are words that turn these open sentences into true or false statements. There are two main types:

1. The Universal Quantifier (∀): Represented by words like 'For all', 'For every'. This quantifier claims that a property is true for every element in a particular domain. For example, the statement 'For every real number x, x² ≥ 0' uses a universal quantifier. This is a true statement.

1. The Existential Quantifier (∃): Represented by words like 'There exists', 'For some', 'There is at least one'. This quantifier claims that there is at least one element in the domain for which a property is true. For example, 'There exists a natural number n such that n + 4 = 7' is a true statement because n=3 works.

Statements containing quantifiers are called quantified statements. A key skill is negating them. The negation of a 'For all' statement is a 'There exists' statement, and vice versa.

• The negation of "For all x, P(x) is true" is "There exists an x such that P(x) is false."
• The negation of "There exists an x such that P(x) is true" is "For all x, P(x) is false."

Key Definitions: Implication, Converse, and Contrapositive

Implication (Conditional Statement)

A statement of the form "if p, then q", denoted as p → q. Here, 'p' is the hypothesis and 'q' is the conclusion. The statement p → q is only considered false when p is true and q is false. In all other cases, it is true.

Converse

The converse of the implication 'if p, then q' is the statement 'if q, then p', denoted as q → p. The truth of an implication does not guarantee the truth of its converse.

Contrapositive

The contrapositive of the implication 'if p, then q' is 'if not q, then not p', denoted as ~q → ~p. An implication and its contrapositive are logically equivalent, meaning they are always either both true or both false.

Worked Examples: Writing Converse and Contrapositive

• Statement: 'If a number n is even, then n² is even.' Write its converse and contrapositive. Step 1: Identify the component statements. p: A number n is even. q: n² is even. The statement is in the form 'if p, then q'. Step 2: Write the Converse (if q, then p). Simply swap p and q. Converse: 'If a number n² is even, then n is even.' Step 3: Write the Contrapositive (if not q, then not p). First, find the negations. ~q: n² is not even (i.e., n² is odd). ~p: A number n is not even (i.e., n is odd). Now form the statement 'if ~q, then ~p'. * Contrapositive: 'If a number n² is odd, then n is odd.' Analysis: In this case, the original statement, its converse, and its contrapositive are all true.
• Statement: 'If a triangle is equilateral, then it is isosceles.' Write its converse and contrapositive. Step 1: Identify the component statements. p: A triangle is equilateral. q: A triangle is isosceles. Step 2: Write the Converse (if q, then p). Converse: 'If a triangle is isosceles, then it is equilateral.' Step 3: Write the Contrapositive (if not q, then not p). ~q: A triangle is not isosceles. ~p: A triangle is not equilateral. Contrapositive: 'If a triangle is not isosceles, then it is not equilateral.' Analysis: The original statement and its contrapositive are true. However, the converse is false (an isosceles triangle does not have to be equilateral).

Exam Tip: Don't Confuse Converse and Contrapositive

A very common mistake in exams is to mix up the converse and the contrapositive. Remember this key difference:

• Converse: Flips the hypothesis and conclusion (p → q becomes q → p). Its truth value can be different from the original statement.
• Contrapositive: Flips AND negates both (p → q becomes ~q → ~p). Its truth value is ALWAYS the same as the original statement. This is called logical equivalence.

In proofs, you can always substitute a statement with its contrapositive. You CANNOT automatically substitute it with its converse.

Example: 'If it is raining, the ground is wet' (True).
The converse 'If the ground is wet, it is raining' is not always true (someone could have used a sprinkler).
The contrapositive 'If the ground is not wet, it is not raining' is true.

Practice Questions with Solutions

• Q: Write the contrapositive and converse of the statement: 'If x is a prime number, then x is odd.' A: Step 1: Identify the component statements 'p' and 'q'. p: x is a prime number. q: x is odd. Step 2: To find the converse, swap p and q to get 'if q, then p'. Converse: 'If x is odd, then x is a prime number.' Step 3: To find the contrapositive, form the statement 'if not q, then not p'. ~q: x is not odd (i.e., x is even). ~p: x is not a prime number. Contrapositive: 'If x is even, then x is not a prime number.' Final Answer: Converse: 'If x is odd, then x is a prime number.' Contrapositive: 'If x is even, then x is not a prime number.' (Note: The original statement and its converse are false because of the prime number 2, but the contrapositive is also false, demonstrating logical equivalence).
• Q: Write the contrapositive and converse of the statement: 'If two lines are parallel, then they do not intersect in the same plane.' A: Step 1: Identify p and q. p: Two lines are parallel. q: They do not intersect in the same plane. Step 2: Write the converse (if q, then p). Converse: 'If two lines do not intersect in the same plane, then they are parallel.' (Note: This is not always true, they could be skew lines). Step 3: Write the contrapositive (if ~q, then ~p). ~q: Two lines intersect in the same plane. ~p: Two lines are not parallel. Contrapositive: 'If two lines intersect in the same plane, then they are not parallel.' Final Answer: Converse: 'If two lines do not intersect in the same plane, then they are parallel.' Contrapositive: 'If two lines intersect in the same plane, then they are not parallel.'
• Q: Write the negation of the statement: 'For every positive real number x, the number x - 1 is also positive.' A: Step 1: Identify the quantifier and the core statement (P(x)). Quantifier: 'For every' (Universal Quantifier). P(x): The number x - 1 is positive. Step 2: Recall the rule for negating a 'For every' statement: 'For every x, P(x)' negates to 'There exists an x such that not P(x)'. Step 3: Apply the rule. The negation of 'x - 1 is positive' is 'x - 1 is not positive' (i.e., x - 1 ≤ 0). Final Answer: 'There exists a positive real number x such that x - 1 is not positive.' or equivalently, 'There exists a positive real number x such that x - 1 ≤ 0.' (For example, x=0.5).
• Q: Given the statement: 'Something is cold implies that it has low temperature.' Write the converse and contrapositive. A: Step 1: Rephrase the statement in the standard 'if-then' format. 'Something is cold implies that it has low temperature' means 'If something is cold, then it has low temperature.' p: Something is cold. q: It has low temperature. Step 2: Write the converse (if q, then p). Converse: 'If something has low temperature, then it is cold.' Step 3: Write the contrapositive (if ~q, then ~p). ~q: It does not have low temperature. ~p: Something is not cold. Contrapositive: 'If something does not have low temperature, then it is not cold.' Final Answer: Converse: 'If something has low temperature, then it is cold.' Contrapositive: 'If something does not have low temperature, then it is not cold.'

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