Logical Disjunction Logical and Operation Conjunction Symbol Conjunction vs Disjunction Conjunction Symbol Math Define Conjunction in Math Conjunction Truth Table Conjunction in Geometry
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And is usually expressed with the prefix operator K, or an infix operator. In mathematics and logic, the infix operator is usually ∧; in electronics ; and in programming languages, & or and. Some programming languages have a related control structure, the short-circuit and, written &&, and then, etc.
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 :
or in logical operator notation:
Here is an example of an argument that fits the form conjunction introduction:
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)
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.
The logical conjunction and in logic is related to, but not the same as, the grammatical conjunction and in natural languages.
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.