Definition:Operation/Binary Operation
Contents |
Definition
A binary operation is the special case of an operation where the operation has exactly two operands.
A binary operation is a mapping $\circ$ from the Cartesian product of two sets $S \times T$ to a universe $\mathbb U$:
- $\circ: S \times T \to \mathbb U: \circ \left ({s, t}\right) = y \in \mathbb U$
If $S = T$, then $\circ$ can be referred to as a binary operation on $S$.
Note that a binary operation is a special case of a general operator, i.e. one that has two operands.
If $\circ$ is a binary operation on $S$, then for any $T \subseteq S$, $\circ \left ({x, y}\right)$ is defined for every $x, y \in T$. So $\circ$ is a binary operation on every $T \subseteq S$.
Infix Notation
A far more common alternative to the notation $\circ \left ({x, y}\right) = z$, which works for a binary operation, is to put the symbol for the operation between the two operands: $z = x \circ y$.
This is called infix notation.
Product
For a given operation $\circ$, let $z = x \circ y$.
Then $z$ is called the product of $x$ and $y$.
This is an extension of the normal definition of product that is encountered in conventional arithmetic.
Nomenclature
Some authors use the term (binary) composition or law of composition for (binary) operation.
Most authors use $\circ$ for composition of relations (which, if you think about it, is itself an operation) as well as for a general operation. To avoid confusion, some authors use $\bullet$ for composition of relations to avoid ambiguity.
Some authors use $\intercal$ (or a variant) called truc (pronounced trook, French for trick or technique
References
- ↑ T.S. Blyth: Set Theory and Abstract Algebra (1975):
The symbol $\intercal$ is called truc ("trook") and is French for "thingummyjig"! The idea it conveys is that what we call our law of composition does not matter, for what we are really interested in are sets of objects and mappings between them.
Sources
- Iain T. Adamson: Introduction to Field Theory (1964)... (next): $\S 1.1$
- W.E. Deskins: Abstract Algebra (1964): $\S 1.4$: Definition $1.10$, Exercise $1.4: 7$
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.1$
- Seth Warner: Modern Algebra (1965): $\S 2$
- Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.5$
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.2$
- B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.1$
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 11$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 27$
- H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.8$
