Day 04Part16(B) – Propositional logic in discrete mathematics – Statements and arguments concept.
Day 04Part16(B) – Propositional logic in discrete mathematics – Statements and arguments concept.
Contents [hide]
- 1 Propositional Logic in Discrete Mathematics – Statements & Arguments
- 2 1. What is a Proposition?
- 3 2. Logical Connectives in Propositional Logic
- 4 3. Arguments in Propositional Logic
- 5 4. Types of Logical Arguments
- 6 5. Valid vs. Invalid Arguments
- 7 6. Truth Tables for Arguments
- 8 7. Applications of Propositional Logic
- 9 Day 04Part16(B) – Propositional logic in discrete mathematics – Statements and arguments concept.
- 10 UNIT – I–DISCRETE MATHEMATICS – SMT1304
- 11 Propositional Logic | Discrete Mathematics
Propositional Logic in Discrete Mathematics – Statements & Arguments
Propositional Logic is a branch of Discrete Mathematics that deals with statements (propositions) and their logical relationships. It helps in mathematical reasoning, computer science, and logic circuits.
1. What is a Proposition?
A proposition is a declarative sentence that is either true (T) or false (F) but not both.
Examples:
“2 + 2 = 4” (True)
“The sky is green.” (False)
Not a Proposition:
“What is your name?” (Not declarative)
“x + 5 = 10” (Depends on x’s value)
2. Logical Connectives in Propositional Logic
We combine propositions using logical connectives:
| Symbol | Name | Example | Meaning |
|---|---|---|---|
| ¬P | Negation | ¬(P: “It is raining”) → “It is NOT raining” | Reverses the truth value |
| P ∧ Q | Conjunction | “I study AND I pass” | Both must be true |
| P ∨ Q | Disjunction | “I study OR I pass” | At least one must be true |
| P → Q | Implication | “If I study, then I pass” | If P is true, then Q must be true |
| P ↔ Q | Biconditional | “I pass if and only if I study” | Both must have the same truth value |
3. Arguments in Propositional Logic
An argument is a sequence of statements (propositions) where some statements (premises) lead to a conclusion.
Example:
Premise 1: If it rains, the ground is wet → (P → Q)
Premise 2: It is raining → (P is True)
Conclusion: The ground is wet → (Q is True)
This is a valid argument (Modus Ponens rule).
4. Types of Logical Arguments
Modus Ponens (Direct Reasoning)
- If P → Q and P is true, then Q is true.
- Example: “If you study, you pass. You studied → So, you passed.”
Modus Tollens (Indirect Reasoning)
- If P → Q and Q is false, then P is false.
- Example: “If it rains, the ground is wet. The ground is not wet → So, it did not rain.”
Hypothetical Syllogism
- If P → Q and Q → R, then P → R.
- Example: “If I study, I pass. If I pass, I graduate → So, if I study, I graduate.”
Disjunctive Syllogism
- If P ∨ Q and ¬P is true, then Q is true.
- Example: “I will either study or play. I did not study → So, I played.”
5. Valid vs. Invalid Arguments
Valid Argument: The conclusion follows logically from the premises.
Invalid Argument (Fallacy): The conclusion does not logically follow.
Example of an Invalid Argument (Fallacy)
Premise: “If you are in India, you are in Asia.” (True)
Premise: “You are in Asia.” (True)
Wrong Conclusion: “You are in India.” (Not necessarily true)
Fallacy Name: Affirming the Consequent
6. Truth Tables for Arguments
We can verify logical validity using truth tables.
Example: Truth Table for P → Q
| P | Q | P → Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Only false when P is True and Q is False.
7. Applications of Propositional Logic
Mathematical Proofs
Artificial Intelligence & Expert Systems
Programming (Conditional Statements)
Logic Circuits (AND, OR, NOT gates)
Cybersecurity (Logical Access Controls)
Would you like some practice questions with solutions to strengthen your understanding?