Day 04Part16(B) – Propositional logic in discrete mathematics – Statements and arguments concept.

Day 04Part16(B) – Propositional logic in discrete mathematics – Statements and arguments concept.

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 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:

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 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

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?

Day 04Part16(B) – Propositional logic in discrete mathematics – Statements and arguments concept.

UNIT – I–DISCRETE MATHEMATICS – SMT1304

Propositional Logic | Discrete Mathematics

Here’s a clear and complete explanation of:

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Day 04 Part 16(B) – Propositional Logic in Discrete Mathematics

Topic: Statements and Arguments – Concept & Explanation

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1. What is Propositional Logic?

Propositional logic (also known as statement logic) deals with propositions (statements) and their logical relationships.

• A proposition is a declarative sentence that is either true or false, but not both.

Examples of valid propositions:

• “The sky is blue.”
• “5 + 3 = 8”
• “Paris is the capital of France.”

Not a proposition:

• “What is your name?” (it’s a question)
• “Please close the door.” (it’s a command)

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2. Statement Logic – Basic Connectives

Symbol	Name	Meaning	Example
¬P	Negation	“not P”	¬(It is raining)
P ∧ Q	Conjunction	“P and Q”	It is hot ∧ humid
P ∨ Q	Disjunction	“P or Q”	I study ∨ I play
P → Q	Implication	“If P, then Q”	If I study → I pass
P Q	Bi-implication	“P if and only if Q”	P Q

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3. What is an Argument in Logic?

An argument is a set of premises (statements) that lead to a conclusion.

• Premises are assumed to be true.
• The conclusion is what we infer from the premises.

Example Argument:

Premise 1: If it rains, the ground gets wet. (P → Q)
Premise 2: It is raining. (P)

Conclusion: The ground gets wet. (Q)

This is a valid argument using Modus Ponens.

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4. Common Inference Rules (Argument Patterns)

Name	Form	Meaning
Modus Ponens	P → Q, P ⟹ Q	If P implies Q and P is true, Q is true
Modus Tollens	P → Q, ¬Q ⟹ ¬P	If P implies Q and Q is false, P is false
Disjunctive Syllogism	P ∨ Q, ¬P ⟹ Q	Either P or Q is true, and P is false, so Q is true
Hypothetical Syllogism	P → Q, Q → R ⟹ P → R	Chain rule of implication

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5. How to Check Argument Validity?

An argument is valid if:

• Whenever all premises are true, the conclusion is also true.

It is invalid if:

• It is possible for premises to be true but conclusion to be false.

You can check validity using:

• Truth tables
• Rules of inference
• Contradictions / counterexamples

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6. Truth Table for Implication (P → Q)

P	Q	P → Q
T	T	T
T	F	F
F	T	T
F	F	T

Implication is only false when P is true and Q is false.

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Summary Points:

• A statement is either true or false.
• Arguments use logical connectives to infer conclusions.
• Use rules of inference (e.g., Modus Ponens) to build or validate arguments.
• Use truth tables to check validity logically.

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Would you like:

• Practice questions with answers?
• A PDF chart of all logic symbols and rules?
• Video lecture or animated explanation?

Let me know how you’d like to continue your learning!

Day 04Part16(B) – Propositional logic in discrete mathematics – Statements and arguments concept.

Lecture 1: Propositional Logic