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 one-to-one functions commute with their inverses.
See also
Associative
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