Logical conjunction

Venn 0000 0001

Variadic logical AND

Venn0001

Venn0001

Venn 0101 0101

Venn 0000 0011

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