In logic and mathematics, and is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true. The logical connective that represents this operator is typically written as or .
"A and B" is true only if A is true and B is true.
An operand of a conjunction is a conjunct.
Related concepts in other fields are:
- In natural language, the coordinating conjunction "and".
- In programming languages, the short-circuit and control structure.
- In set theory, intersection.
- In predicate logic, universal quantification.
And is usually expressed with an infix operator: in mathematics and logic, ∧; in electronics, ; and in programming languages, & or and. In Jan Łukasiewicz's prefix notation for logic, the operator is K, for Polish koniunkcja.
The conjunctive identity is 1, which is to say that AND-ing an expression with 1 will never change the value of the expression. In keeping with the concept of vacuous truth, when conjunction is defined as an operator or function of arbitrary arity, the empty conjunction (AND-ing over an empty set of operands) is often defined as having the result 1.
The truth table of :
Introduction and elimination rules
- Therefore, A and B.
or in logical operator notation:
Here is an example of an argument that fits the form conjunction introduction:
- Bob likes apples.
- Bob likes oranges.
- Therefore, Bob likes apples and oranges.
- A and B.
- Therefore, A.
- A and B.
- Therefore, B.
In logical operator notation:
with exclusive or:
with material nonimplication:
When all inputs are true, the output is true.
|(to be tested)|
When all inputs are false, the output is false.
|(to be tested)|
Walsh spectrum: (1,-1,-1,1)
Applications in computer engineering
In high-level computer programming and digital electronics, logical conjunction is commonly represented by an infix operator, usually as a keyword such as "
AND", an algebraic multiplication, or the ampersand symbol "
&". Many languages also provide short-circuit control structures corresponding to logical conjunction.
Logical conjunction is often used for bitwise operations, where
0 corresponds to false and
1 to true:
0 AND 0=
0 AND 1=
1 AND 0=
1 AND 1=
11000110 AND 10100011=
This can be used to select part of a bitstring using a bit mask. For example,
10011101 AND 00001000 =
00001000 extracts the fifth bit of an 8-bit bitstring.
The membership of an element of an intersection set in set theory is defined in terms of a logical conjunction: x ∈ A ∩ B if and only if (x ∈ A) ∧ (x ∈ B). Through this correspondence, set-theoretic intersection shares several properties with logical conjunction, such as associativity, commutativity, and idempotence.
English "and" has properties not captured by logical conjunction. For example, "and" sometimes implies order. For example, "They got married and had a child" in common discourse means that the marriage came before the child. The word "and" can also imply a partition of a thing into parts, as "The American flag is red, white, and blue." Here it is not meant that the flag is at once red, white, and blue, but rather that it has a part of each color.
- And-inverter graph
- AND gate
- Binary and
- Bitwise AND
- Boolean algebra (logic)
- Boolean algebra topics
- Boolean conjunctive query
- Boolean domain
- Boolean function
- Boolean-valued function
- Conjunction introduction
- Conjunction elimination
- De Morgan's laws
- First-order logic
- Fréchet inequalities
- Grammatical conjunction
- Logical disjunction
- Logical negation
- Logical graph
- Logical value
- Peano-Russell notation
- Propositional calculus
- Józef Maria Bocheński (1959), A Précis of Mathematical Logic, translated by Otto Bird from the French and German editions, Dordrecht, North Holland: D. Reidel, passim.
- Hazewinkel, Michiel, ed. (2001), "Conjunction", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Wolfram MathWorld: Conjunction