In logic and mathematics, or is the truth-functional operator of disjunction, also known as alternation; the or of a set of operands is true if and only ifone or more of its operands is true. The logical connective that represents this operator is typically written as ∨ or +. is true if is true, or if is true, or if both and are true. In logic, or by itself means the inclusiveor, distinguished from an exclusive or, which is false when both of its arguments are true, while an "or" is true in that case. An operand of a disjunction is called a disjunct. Related concepts in other fields are:
Or is usually expressed with an infix operator: in mathematics and logic, ∨; in electronics, +; and in most programming languages, |, ||, or or. In Jan Łukasiewicz's prefix notation for logic, the operator is A, for Polish alternatywa.
Definition
Logical disjunction is an operation on two logical values, typically the values of two propositions, that has a value of false if and only if both of its operands are false. More generally, a disjunction is a logical formula that can have one or more literals separated only by 'or's. A single literal is often considered to be a degenerate disjunction. The disjunctive identity is false, which is to say that the or of an expression with false has the same value as the original expression. In keeping with the concept of vacuous truth, when disjunction is defined as an operator or function of arbitrary arity, the empty disjunction is generally defined as false.
truth-preserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of disjunction.
falsehood-preserving: The interpretation under which all variables are assigned a truth value of 'false' produces a truth value of 'false' as a result of disjunction.
Symbol
The mathematical symbol for logical disjunction varies in the literature. In addition to the word "or", and the formula "Apq", the symbol "", deriving from the Latin word is commonly used for disjunction. For example: "AB " is read as "A or B ". Such a disjunction is false if both A and B are false. In all other cases it is true. All of the following are disjunctions: The corresponding operation in set theory is the set-theoretic union.
The or operator can be used to set bits in a bit field to 1, by or-ing the field with a constant field with the relevant bits set to 1. For example, x = x | 0b00000001 will force the final bit to 1 while leaving other bits unchanged.
Logical operation
Many languages distinguish between bitwise and logical disjunction by providing two distinct operators; in languages following C, bitwise disjunction is performed with the single pipe and logical disjunction with the double pipe operators. Logical disjunction is usually short-circuited; that is, if the first operand evaluates to true then the second operand is not evaluated. The logical disjunction operator thus usually constitutes a sequence point. In a parallel language, it is possible to short-circuit both sides: they are evaluated in parallel, and if one terminates with value true, the other is interrupted. This operator is thus called the parallel or. Although in most languages the type of a logical disjunction expression is boolean and thus can only have the value true or false, in some the logical disjunction operator returns one of its operands: the first operand if it evaluates to a true value, and the second operand otherwise.
As with other notions formalized in mathematical logic, the meaning of the natural-language coordinating conjunction or is closely related to but different from the logical or. For example, "Please ring me or send an email" likely means "do one or the other, but not both". On the other hand, "Her grades are so good that either she's very bright or she studies hard" does not exclude the possibility of both. In other words, in ordinary language "or" can mean either the inclusive "or" or the exclusive "or."
