📘 Chapter 8: Logic – Review Exercise (Solved)
Prepared by Muhammad Tayyab, Subject Specialist Mathematics, Govt Christian High School Daska. Based on PECTAA 2026 syllabus (National Curriculum 2023).
📖 What's Inside: This review exercise covers inductive & deductive reasoning, logical statements, truth tables, conditionals (converse, inverse, contrapositive), axioms, conjectures, theorems, and algebraic proofs with step-by-step logic. Perfect for Punjab Boards exam preparation.
📝 MCQs (Solved with Explanations)
1 Inductive reasoning is based on?
Answer: A) based on repeated experiments – Inductive reasoning draws general conclusions from repeated patterns.
2 Which describes deductive reasoning?
Answer: D) draw conclusion from well-known facts – Deduction starts from general truths.
3 True statement among:
Answer: C) \( \frac{22}{7} \notin Q' \) – \( \frac{22}{7} \) is rational, not irrational.
4 Negation of "The stove is burning"
Answer: A) the stove is not burning
5 Conjunction \( p \land q \) true when?
Answer: B) both p and q are true
6 Conditional false only when?
Answer: A) antecedent true and consequent false
7 Contrapositive of \( q \rightarrow p \) is?
Answer: D) \( \sim p \rightarrow \sim q \)
8 "Every integer > 2 is sum of two primes" is?
Answer: B) conjecture (Goldbach's Conjecture)
9 "A straight line can be drawn between any two points" is?
Answer: C) axiom (Euclidean postulate)
10 "Sum of interior angles of triangle is 180°" is?
Answer: B) theorem (proven statement)
🔄 Converse, Inverse & Contrapositive
(i) \( \sim p \rightarrow q \)
Converse: \( q \rightarrow \sim p \)
Inverse: \( p \rightarrow \sim q \)
Contrapositive: \( \sim q \rightarrow p \)
Converse: \( q \rightarrow \sim p \)
Inverse: \( p \rightarrow \sim q \)
Contrapositive: \( \sim q \rightarrow p \)
(ii) \( q \rightarrow p \)
Converse: \( p \rightarrow q \)
Inverse: \( \sim q \rightarrow \sim p \)
Contrapositive: \( \sim p \rightarrow \sim q \)
Converse: \( p \rightarrow q \)
Inverse: \( \sim q \rightarrow \sim p \)
Contrapositive: \( \sim p \rightarrow \sim q \)
(iii) \( \sim p \rightarrow \sim q \)
Converse: \( \sim q \rightarrow \sim p \)
Inverse: \( p \rightarrow q \)
Contrapositive: \( q \rightarrow p \)
Converse: \( \sim q \rightarrow \sim p \)
Inverse: \( p \rightarrow q \)
Contrapositive: \( q \rightarrow p \)
(iv) \( \sim q \rightarrow \sim p \)
Converse: \( \sim p \rightarrow \sim q \)
Inverse: \( q \rightarrow p \)
Contrapositive: \( p \rightarrow q \)
Converse: \( \sim p \rightarrow \sim q \)
Inverse: \( q \rightarrow p \)
Contrapositive: \( p \rightarrow q \)
📊 Truth Tables
(i) \( \sim(p \vee q) \vee (\sim q) \)
| p | q | \(p \vee q\) | \(\sim(p \vee q)\) | \(\sim q\) | Result |
|---|---|---|---|---|---|
| T | T | T | F | F | F |
| T | F | T | F | T | T |
| F | T | T | F | F | F |
| F | F | F | T | T | T |
(ii) \( \sim(\sim q \vee \sim p) \)
| p | q | \(\sim p\) | \(\sim q\) | \(\sim q \vee \sim p\) | Result |
|---|---|---|---|---|---|
| T | T | F | F | F | T |
| T | F | F | T | T | F |
| F | T | T | F | T | F |
| F | F | T | T | T | F |
📖 Key Definitions (Logic & Reasoning)
Logic: Systematic method of reasoning to interpret truth and deduce new info.
Inductive Reasoning: General conclusion from repeated observations.
Deductive Reasoning: Conclusion from known facts/premises.
Statement: Sentence that is true or false (not both).
Negation (~p): Opposite truth value of p.
Conjunction (p∧q): True only if both true.
Disjunction (p∨q): True if at least one true.
Conditional (p→q): False only when p true & q false.
Axiom/Postulate: Statement accepted without proof.
Theorem: Statement proven using axioms/logic.
Conjecture: Believed true but unproven (e.g., Goldbach's).
Goldbach's Conjecture: Every even > 2 = sum of two primes.
Chapter 1
Real Numbers
Chapter 2
Logarithms
Chapter 3
Sets and Functions
Chapter 4
Factorization and Algebraic Manipulation
Chapter 5
Linear Equations and Inequalities
Chapter 6
Trigonometry
Chapter 7
Coordinate Geometry
Chapter 8
Logic
Chapter 9
Similar Figures
Chapter 10
Graphs of Functions
Chapter 11
Loci and Construction
Chapter 12
Information Handling
Chapter 13
Probability