Which of the following is a statement?

The sum of 2 and 3 is 5.. It is a declarative sentence that is true.

Which of the following is NOT a statement?

Close the window.. An imperative sentence is not a statement.

p ∧ q is true only when:

Both p and q are true. Conjunction needs both parts true.

p ∨ q is false only when:

Both p and q are false. Disjunction is true if at least one part is true.

p → q is false only when:

p is true and q is false. A true hypothesis cannot lead to a false conclusion.

Maharashtra State Board · HSC Commerce · Class 12 (Std XII)

Class 12 Maths and Statistics Part 1 Chapter 1: Mathematical Logic

Short notes, key concepts, quick revision and practice MCQs with answers

Mathematical logic gives precise rules for reasoning. The chapter defines statements, joins them with logical connectives, uses quantifiers, builds truth tables for statement patterns, identifies tautologies, contradictions and contingencies, applies the algebra of statements and represents statements with Venn diagrams.

Key concepts: Mathematical Logic

Statement: A declarative sentence that is either true or false but not both. Imperative, exclamatory and interrogative sentences and open sentences (like x + 5 < 14) are not statements.
Logical connectives: Conjunction p ∧ q (and): true only when both are true. Disjunction p ∨ q (or): false only when both are false. Negation ~p reverses the truth value.
Conditional and biconditional: p → q (if p then q) is false only when p is true and q is false. p ↔ q (p if and only if q) is true when p and q have the same truth value.
Converse, inverse, contrapositive: For p → q: converse q → p, inverse ~p → ~q, contrapositive ~q → ~p. A conditional is equivalent to its contrapositive, and the converse is equivalent to the inverse.
Quantifiers: ∀ (for all) is the universal quantifier and ∃ (there exists) the existential quantifier. The negation of ∀x P(x) is ∃x such that ~P(x), and vice versa.
Tautology, contradiction, contingency: A statement pattern that is always true is a tautology (e.g. p ∨ ~p); always false is a contradiction (e.g. p ∧ ~p); otherwise it is a contingency.
Algebra of statements: Idempotent, commutative, associative, distributive, identity, complement and De Morgan's laws: ~(p ∧ q) ≡ ~p ∨ ~q and ~(p ∨ q) ≡ ~p ∧ ~q. Also p → q ≡ ~p ∨ q and ~(p → q) ≡ p ∧ ~q.
Venn diagrams: Statements about sets such as 'All A are B' (A ⊂ B), 'No A is B' (A ∩ B = ∅) and 'Some A are B' (A ∩ B ≠ ∅) are shown with Venn diagrams.

Quick revision: important points

A truth table for n simple statements has 2ⁿ rows.
~(~p) ≡ p.
p ↔ q ≡ (p → q) ∧ (q → p).
Negation of 'All students are honest' is 'Some students are not honest'.
Statements are denoted by p, q, r; truth values by T and F.

Practice MCQs with answers

Which of the following is a statement?
1. Please open the door.
2. The sum of 2 and 3 is 5.
3. How are you?
4. Wow, what a view!
Show answer
Answer: B. The sum of 2 and 3 is 5.
It is a declarative sentence that is true.
Which of the following is NOT a statement?
1. Mumbai is in Maharashtra.
2. 2 + 2 = 5
3. Close the window.
4. Every square is a rectangle.
Show answer
Answer: C. Close the window.
An imperative sentence is not a statement.
p ∧ q is true only when:
1. p is true
2. q is true
3. Both p and q are true
4. At least one is true
Show answer
Answer: C. Both p and q are true
Conjunction needs both parts true.
p ∨ q is false only when:
1. Both p and q are false
2. p is false
3. q is false
4. Both are true
Show answer
Answer: A. Both p and q are false
Disjunction is true if at least one part is true.
p → q is false only when:
1. p is false and q is true
2. p is true and q is false
3. Both are false
4. Both are true
Show answer
Answer: B. p is true and q is false
A true hypothesis cannot lead to a false conclusion.

All Class 12 Maths and Statistics chapters

Class 12 Maths and Statistics notes: all chapters