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Logical conjunction (AND)

Logical conjunction, commonly called AND, is a fundamental binary operation of logic that returns true only when all its operands are true. It has algebraic properties, digital-circuit equivalents, and everyday uses.

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Logical conjunction is the binary operation in formal logic usually expressed by the word "and" or the symbol ∧. It combines two propositions or truth-bearing statements and yields a single proposition that is true only when both inputs are true. In everyday language conjunction links conditions or facts, but in mathematics and computer science it has a precise truth-functional behaviour.

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Definition and truth table

Formally, conjunction is a function on two truth values that produces a truth value as output. The standard four-case truth table for binary conjunction is:

  1. True ∧ True = True
  2. True ∧ False = False
  3. False ∧ True = False
  4. False ∧ False = False

Because it requires all inputs to be true, conjunction is often called an "all" or "every" operator when generalized to more than two operands.

Algebraic properties

  • Commutativity: A ∧ B = B ∧ A — order does not matter.
  • Associativity: (A ∧ B) ∧ C = A ∧ (B ∧ C) — grouping does not matter.
  • Idempotence: A ∧ A = A — repeating does not change the value.
  • Identity element: True acts as an identity: A ∧ True = A.
  • Annihilator: False annihilates: A ∧ False = False.
  • Distributivity: Conjunction distributes over disjunction and is dual to disjunction under negation (De Morgan's laws).

Variants and distinctions

Conjunction is an inclusive operator: when applied to multiple statements it requires all to be true. This contrasts with exclusive operations such as XOR, which require exactly one true input. In Boolean algebra and programming languages conjunction is implemented by several symbols (∧, &, &&, ·) with subtle differences in evaluation strategy: some forms evaluate both operands always, while short-circuit operators stop once the result is determined.

Uses and examples

Conjunction appears across disciplines. In digital electronics an AND gate realizes the operation with voltage levels, producing a high output only if all inputs are high. In programming, conditional statements use conjunction to require multiple criteria before executing code. In mathematics it combines predicates to form compound assertions. In proofs and formal reasoning, conjunction lets one record simultaneous facts: proving A and B yields A ∧ B, and from A ∧ B either conjunct can be inferred.

Because conjunction is simple but expressive, it is a core connective in propositional logic, predicate logic, Boolean algebra, and practical applications such as circuit design, software conditions, and query filters. Its clear truth-functional nature makes it a foundation for building more complex logical expressions and reasoning systems.

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