binary operation
Definition
Noun: 1. A mathematical or logical operation that requires exactly two inputs (operands) to produce a single output (result). It is a fundamental concept in mathematics, computer science, and logic. 2. An operation defined on a set where the rule combines any two elements of the set to produce another element from the same set. This is a core concept in abstract algebra.
Usage and Examples
- In arithmetic: Addition (), subtraction (), multiplication (), and division () are classic examples of a binary operation. For instance, in , the operation takes the two operands and to produce the result .
- In logic and computing: Logical AND (), OR (), and comparison operations like "greater than" () are binary operations. They take two Boolean values (true/false) or numbers and return a single result.
- In set theory: The union () and intersection () of two sets are binary operations on a collection of sets.
Advanced Usage
- Closure: A key property of a binary operation on a set is closure: applying the operation to any two elements in the set always yields a result that is also within the set. For example, adding two integers always results in an integer.
- Commutative and Associative: A binary operation can be commutative (order doesn't matter, e.g., ) or associative (grouping doesn't matter, e.g., ).
Variants and Related Words
- Binary (adj): Relating to, composed of, or involving two things. Often used in "binary system" (base-2 number system) or "binary choice" (a choice between two options).
- Operand (n): An object or quantity on which an operation is performed. In a binary operation, there are exactly two operands.
- Unary Operation (n): An operation with only one operand (e.g., negation: or logical NOT: ).
Synonyms
- Dyadic operation
- Two-place operation
Related Terms and Concepts
- Boolean Algebra: A branch of algebra where variables represent true/false values and operations (like AND, OR) are binary operations.
- Group Theory: A field in abstract algebra that studies algebraic structures defined by a set and a single binary operation satisfying specific properties (closure, associativity, identity element, inverses).
Noun
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an operation that follows the rules of Boolean algebra; each operand and the result take one of two values
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