Commutative
Operation
Any operation ⊕ for which a⊕b = b⊕a for
all values of a and b. Addition and
multiplication are both commutative. Subtraction, division,
and composition of functions are
not. For example,
5 + 6 = 6 + 5 but 5 – 6 ≠ 6 – 5.
More: Commutativity isn't just a property of
an operation alone. It's actually a property of an operation over
a particular set. For
example, when
we say
addition
is commutative
over the set
of real numbers, we mean that a + b = b + a for
all real numbers a and b. Subtraction is not
commutative over real numbers since we can't say that a – b
= b – a for
all real numbers a and b. Even though a – b
= b – a whenever
a and b are the same, that still
doesn't make subtraction commutative over the set of all real numbers.
Further examples: In this more formal sense,
it is correct to say that matrix
multiplication is not commutative
for square matrices. Even though AB = BA for some square matrices
A and B,
commutativity
does
not
hold
for all square matrices. It is also correct to say composition is
not commutative for functions,
even though onetoone functions commute with their inverses.
See also
Associative
