📒 Definitions (Logic)
Learn and understand Class 9 Mathematics Chapter 8 Definitions on Logic with comprehensive explanations. This page includes free downloadable notes and detailed definitions aligned with the Punjab PECTA 2025 syllabus.
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Chapter 8
Logic
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Chapter 8 - Logic
This chapter focuses on logical reasoning, statements, truth values, and mathematical proofs. Logic is the foundation of mathematical reasoning and problem-solving.
1. What is Logic?
Logic is a systematic method of reasoning that enables us to interpret the meanings of statements, examine their truth, and deduce new information from existing facts. It plays a key role in problem-solving and decision-making.
2. Types of Reasoning
Inductive Reasoning
Inductive reasoning is when we make a general conclusion from repeated observations or experiences.
Example: A person receives a penicillin injection once or twice and experiences a reaction. He concludes that he is allergic to penicillin.
Deductive Reasoning
Deductive reasoning is when we draw a conclusion from already known or accepted facts.
Example: All men are mortal. We are men. So, we are mortal.
3. Statement (Proposition)
A statement is a sentence or mathematical expression that is either true or false, but not both.
4. Logical Operations and Symbols
| Symbol | How to be read | Symbolic Expression | How to be read |
|---|---|---|---|
| ~ | Not | ~p | Not p, negation of p |
| Λ | And | p ∧ q | p and q |
| V | Or | p ∨ q | p or q |
| → | If... then..., implies | p → q | If p then q, p implies q |
| ↔ | if and only if, Is equivalent to | p ↔ q | p if and only if q, p is equivalent to q |
5. Truth Tables
Conjunction (AND) - p ∧ q
| p | q | p ∧ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Disjunction (OR) - p ∨ q
| p | q | p ∨ q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Conditional (Implication) - p → q
| p | q | p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
6. Related Conditionals
For a conditional statement p → q:
- Converse: q → p
- Inverse: ~p → ~q
- Contrapositive: ~q → ~p
Important Note:
A conditional and its contrapositive are logically equivalent. Therefore, any theorem can be proven by proving its contrapositive.
7. Mathematical Proof
In mathematics, a proof is a step-by-step logical explanation that shows a statement is true. It uses definitions, axioms, known theorems, or logical reasoning.
8. Important Concepts
Theorem
A mathematical statement that has been proved to be true using logical steps based on previously accepted facts.
Conjecture
A mathematical statement that is believed to be true based on observations, but not yet proven.
Axiom
A basic mathematical fact that is accepted as true without proof. It forms the foundation of mathematics.
Postulate
Similar to axiom, but used especially in geometry.
9. Famous Theorems and Conjectures
Fundamental Theorem of Arithmetic
Every integer greater than 1 can be uniquely written as a product of prime numbers, ignoring the order of the factors.
Goldbach's Conjecture
Every even number greater than 2 is the sum of two prime numbers.
Examples: 4 = 2 + 2, 6 = 3 + 3, 12 = 5 + 7
Fermat's Last Theorem
There are no positive numbers a, b, c such that aⁿ + bⁿ = cⁿ for any n > 2.
Created by Hira Science Academy | Aligned with PECTA 2025 Syllabus