Logical conjunction

Venn 0000 0001

Variadic logical AND

Venn0001

Venn0001

Venn 0101 0101

Venn 0000 0011

Logical operation that outputs true only when both inputs are true

For other uses, see Conjunction (disambiguation).

Logical conjunction is a fundamental operation in mathematical logic and Boolean algebra. It is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true. The logical conjunction of the propositions A and B is commonly denoted as A ∧ B or A AND B.

Definition

In formal terms, the logical conjunction of two propositions A and B is defined as follows:

• A ∧ B is true if and only if both A is true and B is true.
• A ∧ B is false if either A is false, B is false, or both are false.

Truth Table

The truth table for logical conjunction is as follows:

Truth table for A ∧ B

A	B	A ∧ B
true	true	true
true	false	false
false	true	false
false	false	false

Properties

Logical conjunction has several important properties:

• Commutativity: A ∧ B is equivalent to B ∧ A.
• Associativity: (A ∧ B) ∧ C is equivalent to A ∧ (B ∧ C).
• Idempotence: A ∧ A is equivalent to A.
• Identity element: The identity element for conjunction is true, meaning A ∧ true is equivalent to A.
• Distributivity: Conjunction distributes over disjunction, i.e., A ∧ (B ∨ C) is equivalent to (A ∧ B) ∨ (A ∧ C).

Applications

Logical conjunction is widely used in various fields such as:

• Computer science: In programming languages, logical conjunction is used in conditional statements and loops.
• Digital electronics: In logic gates, the AND gate implements logical conjunction.
• Mathematics: In set theory, the intersection of sets is analogous to logical conjunction.

Related Concepts

• Logical disjunction
• Logical negation
• Boolean algebra
• Truth table
• Propositional logic
• Predicate logic

See Also

• AND gate
• Boolean function
• Logical connective
• Set theory

References

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