Mathematical Reasoning Class 11 Chapter Notes

Key Terms in Mathematical Reasoning

Statement (Proposition)

A declarative sentence which is either true or false, but not both. For example, '2 + 2 = 4' is a true statement.

Simple Statement

A statement whose truth value does not explicitly depend on another statement. It cannot be broken down further.

Compound Statement

A statement formed by combining two or more simple statements using logical connectives like 'and', 'or', 'not', 'if-then'.

Negation (~)

The denial of a statement. The negation of a statement 'p' is 'not p', denoted by ~p.

Tautology

A compound statement which is always true, regardless of the truth values of its component statements.

Contradiction (Fallacy)

A compound statement which is always false, regardless of the truth values of its components.

Converse

For an implication 'if p, then q' (p → q), the converse is 'if q, then p' (q → p).

Inverse

For an implication 'if p, then q' (p → q), the inverse is 'if not p, then not q' (~p → ~q).

Contrapositive

For an implication 'if p, then q' (p → q), the contrapositive is 'if not q, then not p' (~q → ~p). It is logically equivalent to the original statement.

Statements vs. Non-Statements

The first step in mathematical reasoning is to identify a valid mathematical statement. A sentence qualifies as a statement if and only if it has a definite truth value — it must be unambiguously true or false. It cannot be both, nor can it be subjective or dependent on context.

For instance, "The sum of the angles in a triangle is 180 degrees" is a true statement. "The Sun revolves around the Earth" is a false statement. Both are valid mathematical statements because they have a clear truth value.

However, consider the following sentences:

• "Close the door." (This is a command, not a statement).
• "How old are you?" (This is a question, not a statement).
• "Mathematics is an interesting subject." (This is an opinion or subjective sentence, its truth varies from person to person).
• "x + 5 = 10." (This is an open sentence. Its truth depends on the value of 'x'. It becomes a statement once 'x' is defined).

Recognizing this distinction is critical because the rules of logic, connectives, and proofs can only be applied to valid mathematical statements.

Logical Connectives: AND, OR

Must Remember

• An implication (p → q) is logically equivalent to its contrapositive (~q → ~p).
• The converse (q → p) and inverse (~p → ~q) are logically equivalent to each other, but not to the original statement.
• De Morgan's Law 1: The negation of (p and q) is (~p or ~q). ~(p ∧ q) ≡ ~p ∨ ~q.
• De Morgan's Law 2: The negation of (p or q) is (~p and ~q). ~(p ∨ q) ≡ ~p ∧ ~q.
• The 'OR' (∨) connective is inclusive by default in mathematics. It means 'p or q or both'.
• The biconditional statement 'p if and only if q' (p ↔ q) is true only when p and q have the same truth value.
• To prove p ↔ q, you must prove both p → q and q → p.
• To disprove a 'For all' statement, a single counterexample is sufficient.
• Negation of a quantifier: ~(∀x, P(x)) becomes ∃x, ~P(x). And ~(∃x, P(x)) becomes ∀x, ~P(x).

The 'If-Then' Family: Implication and Its Relatives

Methods for Validating Statements

1. —
2. —
3. —
4. —

Worked Mini-Example

• {"title":"Writing the Converse and Contrapositive","bodyMarkdown":"Statement: \"If a triangle is equilateral, then it is isosceles.\"\n\n- p: A triangle is equilateral.\n- q: It is isosceles.\n\nConverse (q → p): \"If a triangle is isosceles, then it is equilateral.\"\n_(This is false, as an isosceles triangle can have only two equal sides)._\n\nContrapositive (~q → ~p): \"If a triangle is not isosceles, then it is not equilateral.\"\n_(This is true, and logically equivalent to the original statement)._"}

Exam Traps and Key Focus Areas

Students often confuse Converse and Contrapositive. A simple trick to remember: Converse = Flip (p → q becomes q → p), Contrapositive = Flip AND Negate (p → q becomes ~q → ~p).

In exams, questions frequently test the negation of compound statements. Always apply De Morgan's laws: ~(p ∧ q) ≡ ~p ∨ ~q and ~(p ∨ q) ≡ ~p ∧ ~q. Don't forget to flip the connective from 'and' to 'or' and vice versa. This is a common source of error.

Quick Revision Check

• Is the sentence 'Tomorrow is Wednesday' a mathematical statement? No. Its truth value depends on the context of 'today', so it is not a mathematically valid statement.
• What is the negation of the statement 'Some students are lazy'? Using quantifiers, this is 'There exists a student who is lazy'. Its negation is 'All students are not lazy' or 'No student is lazy'.
• Write the converse of: 'If a number n is even, then n² is even'. Converse: 'If a number n² is even, then n is even'.
• Is the statement '(p → q) ↔ (~q → ~p)' a tautology or a contradiction? It is a tautology, because a conditional statement is always logically equivalent to its contrapositive.

Frequently Asked Questions

Frequently Asked Questions