In logic, mathematics and linguistics, and ( ∧ {\displaystyle \wedge } ) is the truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as ∧ {\displaystyle \wedge } or & {\displaystyle \&} or K {\displaystyle K} (prefix) or × {\displaystyle \times } or ⋅ {\displaystyle \cdot } in which ∧ {\displaystyle \wedge } is the most modern and widely used. The and of a set of operands is true if and only if all of its operands are true, i.e., A ∧ B {\displaystyle A\land B} is true if and only if A {\displaystyle A} is true and B {\displaystyle B} is true. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields:
In natural language, the denotation of expressions such as English "and"; In programming languages, the short-circuit and control structure; In set theory, intersection. In lattice theory, logical conjunction (greatest lower bound).
Notation And is usually denoted by an infix operator: in mathematics and logic, it is denoted by a "wedge" ∧ {\displaystyle \wedge } (Unicode U+2227 ∧ LOGICAL AND), & {\displaystyle \&} or × {\displaystyle \times } ; in electronics, ⋅ {\displaystyle \cdot } ; and in programming languages &, &&, or and. In Jan Łukasiewicz's prefix notation for logic, the operator is K {\displaystyle K} , for Polish koniunkcja. In mathematics, the conjunction of an arbitrary number of elements a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} can be denoted as an iterated binary operation using a "big wedge" ⋀ (Unicode U+22C0 ⋀ N-ARY LOGICAL AND):
⋀ i = 1 n a i = a 1 ∧ a 2 ∧ … a n − 1 ∧ a n {\displaystyle \bigwedge _{i=1}^{n}a_{i}=a_{1}\wedge a_{2}\wedge \ldots a_{n-1}\wedge a_{n}}
Definition In classical logic, logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if (also known as iff) both of its operands are true. The conjunctive identity is true, which is to say that AND-ing an expression with true 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 true.
Truth table
The truth table of A ∧ B {\displaystyle A\land B} :
Defined by other operators In systems where logical conjunction is not a primitive, it may be defined as
A ∧ B = ¬ ( A → ¬ B ) {\displaystyle A\land B=\neg (A\to \neg B)}
It can be checked by the following truth table (compare the last two columns):
or
A ∧ B = ¬ ( ¬ A ∨ ¬ B ) . {\displaystyle A\land B=\neg (\neg A\lor \neg B).}
It can be checked by the following truth table (compare the last two columns):
Introduction and elimination rules As a rule of inference, conjunction introduction is a classically valid, simple argument form. The argument form has two premises, A {\displaystyle A} and B {\displaystyle B} . Intuitively, it permits the inference of their conjunction.
A {\displaystyle A} ,
B {\displaystyle B} . Therefore, A and B. or in logical operator notation, where ⊢ {\displaystyle \vdash } expresses provability:
⊢ A , {\displaystyle \vdash A,}
⊢ B {\displaystyle \vdash B}
⊢ A ∧ B {\displaystyle \vdash A\land B}
Here is an example of an argument that fits the form conjunction introduction:
Bob likes apples. Bob likes oranges. Therefore, Bob likes apples and Bob likes oranges. Conjunction elimination is another classically valid, simple argument form. Intuitively, it permits the inference from any conjunction of either element of that conjunction.
A {\displaystyle A} and B {\displaystyle B} . Therefore, A {\displaystyle A} . ...or alternatively,
A {\displaystyle A} and B {\displaystyle B} . Therefore, B {\displaystyle B} . In logical operator notation:
⊢ A ∧ B {\displaystyle \vdash A\land B}
⊢ A {\displaystyle \vdash A}
...or alternatively,
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