- Logical conjunction
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 truth table of :
INPUT OUTPUT A B A AND B 0 0 0 0 1 0 1 0 0 1 1 1
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
0corresponds to false and
0 AND 0=
0 AND 1=
1 AND 0=
1 AND 1=
The operation can also be applied to two binary words viewed as bitstrings of equal length, by taking the bitwise AND of each pair of bits at corresponding positions. For example:
11000110 AND 10100011=
This can be used to select part of a bitstring using a bit mask. For example,
10011101 AND 00001000=
00001000extracts the fifth bit of an 8-bit bitstring.
The intersection used in set theory is defined in terms of a logical conjunction: x ∈ A ∩ B if and only if (x ∈ A) ∧ (x ∈ B). Because of this, 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.
- 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
- Grammatical conjunction
- Logical disjunction
- Logical negation
- Logical graph
- Logical value
- Peano-Russell notation
- Propositional calculus
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