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.
Contents [hide]
- 1 Disjunction Operator (∨) in Propositional Logic – Discrete Mathematics for Computer Science
- 2 1. Definition of Disjunction (∨)
- 3 2. Symbolic Representation
- 4 3. Key Properties of Disjunction (∨)
- 5 (A) Commutative Property:
- 6 (B) Associative Property:
- 7 (C) Identity Law:
- 8 (D) Domination Law:
- 9 (E) Idempotent Law:
- 10 (F) Distributive Law:
- 11 4. Application in Computer Science
- 12 5. Disjunction in Logic Gates
- 13 Conclusion
- 14 Day 04Part 06- Discrete mathematics for computer science – Dis-junction Operator of proposition.
- 15 Discrete Mathematics for Computer Science Prof. Raissa D …
- 16 Propositional Logic Discrete Mathematics
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 (∨)
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The disjunction (P ∨ Q) of two propositions P and Q is true if either P is true, Q is true, or both are true.
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The only case when P ∨ Q is false is when both P and Q are false.
Truth Table for Disjunction (P ∨ Q):
| P | Q | P ∨ Q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
2. Symbolic Representation
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P ∨ Q reads as “P OR Q“
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Example:
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P = “It is raining.”
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Q = “It is cloudy.”
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P ∨ Q = “It is raining OR it is cloudy.”
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The statement is true if at least one of the conditions is true.
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3. Key Properties of Disjunction (∨)
(A) Commutative Property:
P∨Q=Q∨PP ∨ Q = Q ∨ P
Example: “It is raining OR it is cloudy” is the same as “It is cloudy OR it is raining.”
(B) Associative Property:
(P∨Q)∨R=P∨(Q∨R)(P ∨ Q) ∨ R = P ∨ (Q ∨ 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∨False=PP ∨ \text{False} = P
Example: “It is sunny OR False” → The truth value remains that of “It is sunny.”
(D) Domination Law:
P∨True=TrueP ∨ \text{True} = \text{True}
Example: “It is sunny OR True” → Always True.
(E) Idempotent Law:
P∨P=PP ∨ P = P
Example: “It is raining OR it is raining” → No change in truth value.
(F) Distributive Law:
P∨(Q∧R)=(P∨Q)∧(P∨R)P ∨ (Q ∧ R) = (P ∨ Q) ∧ (P ∨ R)
4. Application in Computer Science
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Boolean Algebra: Used in circuit design and logic gates.
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Programming: Conditional statements (e.g.,
if (A || B) {}in C, Java, Python). -
Database Queries: SQL uses
ORto 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
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Implemented as an “OR” gate in digital circuits.
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Output is 1 (True) if at least one input is 1.
| A | B | A OR B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Conclusion
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The disjunction operator (∨) is a fundamental part of propositional logic, used extensively in mathematics, computer science, and electronics.
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It follows commutative, associative, and distributive properties.
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It has practical applications in programming, databases, and logic circuit design.
Would you like more examples or practice problems on this topic?