Class 11 Mathematics Notes Chapter 14 (Chapter 14) – Examplar Problems (English) Book

Detailed Notes with MCQs of Chapter 14: Mathematical Reasoning from your NCERT Exemplar. This chapter is fundamental, not just for your Class 11 exams, but it also builds the logical foundation required for various competitive government exams where logical deduction and statement analysis are often tested. Pay close attention to the definitions and rules.

Chapter 14: Mathematical Reasoning - Detailed Notes

1. Introduction to Statements (Propositions)

• Definition: A statement is a declarative sentence which is either definitively true or definitively false, but not both simultaneously. Ambiguous sentences, questions, commands, exclamations, or opinions are not mathematical statements.
  • Example (Statement): "The sum of angles in a triangle is 180°." (True)
  • Example (Statement): "3 + 5 = 7." (False)
  • Example (Not a Statement): "Where are you going?" (Question)
  • Example (Not a Statement): "Mathematics is difficult." (Opinion/Ambiguous)
  • Example (Not a Statement): "x + 2 = 5." (Open sentence - becomes a statement only when x is specified).
• Truth Value: The truth (T) or falsity (F) of a statement is called its truth value.

2. Logical Connectives and Compound Statements

Simple statements can be combined using logical connectives to form compound statements.

• a) Negation (~ or ¬):

  • Meaning: Denies or contradicts a statement. Read as "not p".
  • Rule: If statement 'p' is true, then '~p' is false. If 'p' is false, then '~p' is true.
  • Example: p: "New Delhi is the capital of India." (T)
    ~p: "New Delhi is not the capital of India." (F) OR "It is false that New Delhi is the capital of India." (F)
  • Truth Table:

    p	~p
    T	F
    F	T

• b) Conjunction (∧):

  • Meaning: Connects two statements using "and". Read as "p and q".
  • Rule: The compound statement 'p ∧ q' is true only if both 'p' and 'q' are true. Otherwise, it's false.
  • Example: p: "3 is an odd number." (T), q: "5 is a prime number." (T)
    p ∧ q: "3 is an odd number and 5 is a prime number." (T)
  • Truth Table:

    p	q	p ∧ q
    T	T	T
    T	F	F
    F	T	F
    F	F	F

• c) Disjunction (∨):

  • Meaning: Connects two statements using "or" (inclusive or). Read as "p or q".
  • Rule: The compound statement 'p ∨ q' is false only if both 'p' and 'q' are false. Otherwise, it's true (meaning at least one is true).
  • Example: p: "A square has 4 sides." (T), q: "A triangle has 5 sides." (F)
    p ∨ q: "A square has 4 sides or a triangle has 5 sides." (T)
  • Truth Table:

    p	q	p ∨ q
    T	T	T
    T	F	T
    F	T	T
    F	F	F

• d) Implication (→ or ⇒):

  • Meaning: Represents "if p, then q". 'p' is the hypothesis (antecedent), 'q' is the conclusion (consequent).
  • Rule: The implication 'p → q' is false only if 'p' is true and 'q' is false. In all other cases, it's true. This is crucial – a false hypothesis can imply anything (true or false).
  • Equivalent Phrases: "p implies q", "p is sufficient for q", "q is necessary for p", "p only if q", "q whenever p", "~q implies ~p" (Contrapositive).
  • Example: p: "It is raining." q: "The ground is wet."
    p → q: "If it is raining, then the ground is wet." (This is generally true. If p is T and q is F - i.e., it's raining but the ground isn't wet - the implication is False).
  • Truth Table:

    p	q	p → q
    T	T	T
    T	F	F
    F	T	T
    F	F	T

• e) Biconditional (↔ or ⇔):

  • Meaning: Represents "p if and only if q" (p iff q).
  • Rule: The biconditional 'p ↔ q' is true only if 'p' and 'q' have the same truth value (both true or both false).
  • Equivalence: p ↔ q is logically equivalent to (p → q) ∧ (q → p).
  • Example: p: "A triangle is equilateral." q: "All three angles of a triangle are equal."
    p ↔ q: "A triangle is equilateral if and only if all three angles of the triangle are equal." (T)
  • Truth Table:

    p	q	p ↔ q
    T	T	T
    T	F	F
    F	T	F
    F	F	T

3. Quantifiers

Quantifiers modify statements by specifying the extent to which they apply.

• a) Universal Quantifier (∀):

  • Symbol: ∀
  • Meaning: "For all", "For every", "For each".
  • Example: "∀ x ∈ N, x + 1 > x" (For all natural numbers x, x+1 is greater than x). (True)

• b) Existential Quantifier (∃):

  • Symbol: ∃
  • Meaning: "There exists", "For some", "There is at least one".
  • Example: "∃ x ∈ Z such that x² = 4" (There exists an integer x such that x squared equals 4). (True, x=2 or x=-2).

• c) Negation of Quantified Statements: This is very important for proofs and disproofs.

  • ~(∀x, P(x)) is equivalent to ∃x, ~P(x).
    • Meaning: To negate "For all x, P(x) is true", you need to show "There exists at least one x for which P(x) is false".
    • Example: Negation of "All students like Mathematics" is "There exists at least one student who does not like Mathematics".
  • ~(∃x, P(x)) is equivalent to ∀x, ~P(x).
    • Meaning: To negate "There exists an x for which P(x) is true", you need to show "For all x, P(x) is false".
    • Example: Negation of "Some prime numbers are even" is "All prime numbers are not even" (or "No prime number is even").

4. Tautology and Contradiction

• Tautology: A compound statement that is always true, regardless of the truth values of its component statements. (e.g., p ∨ ~p).
• Contradiction (Fallacy): A compound statement that is always false, regardless of the truth values of its component statements. (e.g., p ∧ ~p).

5. Converse, Inverse, and Contrapositive

Given an implication p → q:

• Converse: q → p
• Inverse: ~p → ~q
• Contrapositive: ~q → ~p

Key Equivalence:

• An implication (p → q) is logically equivalent to its contrapositive (~q → ~p). Their truth tables are identical.
• The converse (q → p) is logically equivalent to the inverse (~p → ~q).

Example: Statement: "If a number is divisible by 4 (p), then it is divisible by 2 (q)." (True)

• Converse: "If a number is divisible by 2 (q), then it is divisible by 4 (p)." (False, e.g., 6)
• Inverse: "If a number is not divisible by 4 (~p), then it is not divisible by 2 (~q)." (False, e.g., 6)
• Contrapositive: "If a number is not divisible by 2 (~q), then it is not divisible by 4 (~p)." (True)

6. Methods of Validating Statements

How to check if a given mathematical statement is true or false.

• Direct Proof (for p → q): Assume 'p' is true and use axioms, definitions, and previously proven theorems to logically deduce that 'q' must also be true.
• Proof by Contrapositive (for p → q): Prove the equivalent statement ~q → ~p. Assume ~q is true and deduce that ~p must be true.
• Proof by Contradiction: To prove 'p' is true, assume '~p' is true and derive a logical contradiction (a statement that is always false, like r ∧ ~r). This shows the assumption '~p' must be false, hence 'p' must be true.
  • To prove p → q using contradiction: Assume p is true AND q is false (i.e., assume the implication is false), then derive a contradiction.
• Proof by Counterexample: To prove a universally quantified statement (∀x, P(x)) is false, you only need to find one specific value of x (a counterexample) for which P(x) is false.
  • Example: To disprove "∀ n ∈ N, n² + n + 41 is prime", find a counterexample. For n=40, 40² + 40 + 41 = 1600 + 40 + 41 = 1681 = 41², which is not prime.
• Validating "And" (p ∧ q): Show both p and q are true individually.
• Validating "Or" (p ∨ q): Show that at least one of p or q is true. (Often done by assuming one is false and showing the other must be true).
• Validating "If and Only If" (p ↔ q): Prove both implications: (i) p → q and (ii) q → p.

Importance for Government Exams:
Questions might involve:

• Identifying valid statements.
• Negating complex statements (especially involving quantifiers or multiple connectives).
• Finding the contrapositive/converse/inverse.
• Drawing conclusions from given premises using logical rules (syllogisms, although not explicitly detailed here, rely on these principles).
• Identifying tautologies/contradictions.

Multiple Choice Questions (MCQs)

1. Which of the following is not a statement in mathematical logic?
   (a) The Earth is a planet.
   (b) 5 is a prime number.
   (c) Turn off the fan.
   (d) √2 is an irrational number.

2. The negation of the statement "∀ x ∈ R, x² ≥ 0" is:
   (a) ∀ x ∈ R, x² < 0
   (b) ∃ x ∈ R such that x² < 0
   (c) ∃ x ∈ R such that x² ≤ 0
   (d) ∀ x ∈ R, x² ≠ 0

3. Consider p: "It is raining" and q: "The match is cancelled". The statement "If it is raining, then the match is cancelled" is represented by:
   (a) p ∧ q
   (b) p ∨ q
   (c) p → q
   (d) p ↔ q

4. The contrapositive of the statement "If x is a prime number, then x is odd" is:
   (a) If x is not odd, then x is not a prime number.
   (b) If x is not a prime number, then x is not odd.
   (c) If x is odd, then x is a prime number.
   (d) x is a prime number and x is not odd.

5. Which of the following compound statements is a tautology?
   (a) p ∧ ~p
   (b) p → (p ∨ q)
   (c) (p ∧ q) → p
   (d) Both (b) and (c)

6. The statement (p → q) ↔ (~q → ~p) is:
   (a) A tautology
   (b) A contradiction
   (c) Equivalent to p ∧ q
   (d) Equivalent to p ∨ q

7. What is the truth value of the statement: "If 4 is an odd number (p), then 10 is divisible by 3 (q)"?
   (a) True
   (b) False
   (c) Cannot be determined
   (d) Both True and False

8. The negation of "Some triangles are equilateral" is:
   (a) All triangles are equilateral.
   (b) No triangle is equilateral.
   (c) Some triangles are not equilateral.
   (d) If a figure is a triangle, then it is not equilateral.

9. Consider the statement: "If n is an integer such that n > 2, then n² > 4". Which of the following is a valid converse to this statement?
   (a) If n² ≤ 4, then n ≤ 2.
   (b) If n² > 4, then n > 2.
   (c) If n ≤ 2, then n² ≤ 4.
   (d) If n² > 4, then n is an integer such that n > 2.

10. To disprove the statement "For every real number x, x² + 1 > 0", one needs to find a real number x such that:
    (a) x² + 1 < 0
    (b) x² + 1 ≤ 0
    (c) x² + 1 = 0
    (d) The statement cannot be disproven.

Answer Key for MCQs:

1. (c) - It's a command.
2. (b) - Negation of ∀ is ∃, negation of P(x) is ~P(x). ~(x² ≥ 0) is x² < 0.
3. (c) - Standard "If p, then q" structure.
4. (a) - Contrapositive is ~q → ~p. p: "x is prime", q: "x is odd". ~q: "x is not odd", ~p: "x is not prime".
5. (d) - Both (b) and (c) are always true. You can verify with truth tables. For (b), if p is T, p∨q is T, so T→T is T. If p is F, F→(F∨q) is T. For (c), if p∧q is T, then p is T, so T→T is T. If p∧q is F, then F→p is T.
6. (a) - An implication is always logically equivalent to its contrapositive. Thus, the biconditional connecting them is always true (a tautology).
7. (a) - The hypothesis p ("4 is an odd number") is False. When the hypothesis is false, the implication p → q is always True, regardless of the truth value of q.
8. (b) - "Some triangles are equilateral" means "∃ triangle T such that T is equilateral". The negation is "∀ triangle T, T is not equilateral", which means "No triangle is equilateral".
9. (d) - The original statement is p → q where p: "n is an integer such that n > 2" and q: "n² > 4". The converse is q → p. Option (d) correctly states "If n² > 4, then n is an integer such that n > 2". Option (b) is close but misses the "n is an integer" condition which was part of the original hypothesis 'p'.
10. (d) - The statement "For every real number x, x² + 1 > 0" is actually true. Since x² ≥ 0 for all real x, x² + 1 ≥ 1, which means x² + 1 is always strictly positive. Therefore, you cannot find a counterexample, and the statement cannot be disproven. (The question asks what one needs to find to disprove it, which would be an x where x²+1 ≤ 0, as per option (b), but such an x doesn't exist in real numbers). Given the options, (d) is the most appropriate conclusion in the context of mathematical validity.

Make sure you understand the reasoning behind each answer, especially the nuances in quantifiers and implications. Good luck with your preparation!