Day 04Part 06- Discrete mathematics for computer science – Dis-junction Operator of proposition.

Day 04Part 06- Discrete mathematics for computer science – Dis-junction Operator of proposition.

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Disjunction Operator (∨) in Propositional Logic – Discrete Mathematics for Computer Science

In propositional logic, the disjunction operator (∨) represents the logical “OR” operation. It is used to form compound statements that are true if at least one of the given propositions is true.

1. Definition of Disjunction (∨)

• The disjunction (P ∨ Q) of two propositions P and Q is true if either P is true, Q is true, or both are true.

• The only case when P ∨ Q is false is when both P and Q are false.

Truth Table for Disjunction (P ∨ Q):

2. Symbolic Representation

• P ∨ Q reads as “P OR Q“

• Example:

  • P = “It is raining.”

  • Q = “It is cloudy.”

  • P ∨ Q = “It is raining OR it is cloudy.”

  • The statement is true if at least one of the conditions is true.

3. Key Properties of Disjunction (∨)

(A) Commutative Property:

$$P\vee Q=Q\vee P$$

 Example: “It is raining OR it is cloudy” is the same as “It is cloudy OR it is raining.”

(B) Associative Property:

$$(P\vee Q)\vee R=P\vee (Q\vee R)$$

 Example: If P = “Hot”, Q = “Cold”, and R = “Windy”, the order in which we group them doesn’t change the truth.

(C) Identity Law:

$$P\vee \text{False}=P$$

 Example: “It is sunny OR False” → The truth value remains that of “It is sunny.”

(D) Domination Law:

$$P\vee \text{True}=\text{True}$$

 Example: “It is sunny OR True” → Always True.

(E) Idempotent Law:

$$P\vee P=P$$

 Example: “It is raining OR it is raining” → No change in truth value.

(F) Distributive Law:

$$P\vee (Q\wedge R)=(P\vee Q)\wedge (P\vee R)$$

4. Application in Computer Science

• Boolean Algebra: Used in circuit design and logic gates.

• Programming: Conditional statements (e.g., if (A || B) {} in C, Java, Python).

• Database Queries: SQL uses OR to filter data (SELECT * FROM students WHERE grade = 'A' OR attendance > 80;).

• Artificial Intelligence & Machine Learning: Logical operations in rule-based systems.

5. Disjunction in Logic Gates

• Implemented as an “OR” gate in digital circuits.

• Output is 1 (True) if at least one input is 1.

Conclusion

• The disjunction operator (∨) is a fundamental part of propositional logic, used extensively in mathematics, computer science, and electronics.

• It follows commutative, associative, and distributive properties.

• It has practical applications in programming, databases, and logic circuit design.

Would you like more examples or practice problems on this topic?

Day 04Part 06- Discrete mathematics for computer science – Dis-junction Operator of proposition.

Discrete Mathematics for Computer Science Prof. Raissa D …

Propositional Logic Discrete Mathematics