Everything about Binary Operations totally explained
In
mathematics, a
binary operation is a calculation involving two
operands, in other words, an operation whose
arity is two. Binary operations can be accomplished using either a
binary function or
binary operator. Binary operations are sometimes called
dyadic operations in order to avoid confusion with the
binary numeral system. Examples include the familiar
arithmetic operations of
addition,
subtraction,
multiplication and
division.
More precisely, a binary operation on a
set S is a
binary relation that maps elements of the
Cartesian product S ×
S to
S:
»
If
f isn't a function, but is instead a
partial function, it's called a
partial operation. For instance, division of real numbers is a partial function, because one can't
divide by zero: 1/0 and 0/0 are not defined.
Sometimes, especially in
computer science, the term is used for any
binary function. That
f takes values in the same set
S that provides its arguments is the property of
closure.
Binary operations are the keystone of algebraic structures studied in
abstract algebra: they form part of
groups,
monoids,
semigroups,
rings, and more.
Most generally, a
magma is a set together with any binary operation defined on it.
Many binary operations of interest in both algebra and formal logic are
commutative or
associative.
Many also have
identity elements and
inverse elements.
Typical examples of binary operations are the
addition (+) and
multiplication (×) of
numbers and
matrices as well as
composition of functions on a single set.
An example of an operation that isn't
commutative is
subtraction (−). Examples of partial operations that are not commutative include
division (/),
exponentiation(^), and
super-exponentiation(↑↑).
Binary operations are often written using
infix notation such as
a ∗
b,
a +
b, or
a ·
b rather than by functional notation of the form
f(
a,
b).
Sometimes they're even written just by :
ab. Powers are usually also written without operator, but with the second argument as
superscript.
Binary operations sometimes use prefix or postfix notation, this dispenses with parentheses. Prefix notation is also called
Polish notation; postfix notation, also called
reverse Polish notation, is probably more often encountered.
Pair and tuple
A binary operation,
ab, depends on the
ordered pair (
a,
b) and so (
ab)
c (where the parentheses here mean first operate on the ordered pair (a, b) and then operate on the result of that using the ordered pair ((ab), c) depends in general on the ordered pair ((
a,
b),
c). Thus, for the general, non-associative case, binary operations can be represented with
binary trees.
However:
- If the operation is associative, (ab)c=a(bc), then the value depends only on the tuple (a,b,c).
- If the operation is commutative, ab=ba, then the value depends only on the multiset .
External binary operations
An
external binary operation is a binary function from
K and
S to
S.
This differs from a binary operation in the strict sense in that
K need not be
S; its elements come from
outside.
An example of an
external binary operation is
scalar multiplication in
linear algebra.
Here
K is a
field and
S is a
vector space over that field.
An
external binary operation may alternatively be viewed as an
action;
K is acting on
S.
Further Information
Get more info on 'Binary Operations'.
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