Commutativity

https://en.wikipedia.org/wiki/Commutative_property

A binary operation is commutative if changing the order of the operands does not change the result.

a ⨀ b = b ⨀ a

$\forall x,y\ .\ x \star y = y \star x$

Integers are commutative:

• under addition: $a+b=b+a$

• under multiplication: $ab=ba$

• under percentage: $(ab=ba) \to (a\%\ of\ b = b\%\ of\ a)$

There are operations (division and subtraction) that are not commutative, referred to as noncommutative operations.

The commutative property (or commutative law) is a property generally associated with binary operations and functions.

If the commutative property holds for a pair of elements under a certain binary operation then the two elements are said to commute under that operation.

Commutativity vs associativity

Commutativity does not imply associativity.

A very important example of an operation that is commutative but non-associative is that of finite-precision floating point numbers under addition:

(a + -a) + b = b, but a + (-a + b) ~ b

may differ from b since the sum -a + b can involve a loss of precision. The latter is especially the case if a and b are nearly equal, then their sum might be rounded to zero (despite that their corresponding real sum is nonzero).

Another example is the binary operation, denoted by ⊙, on the integers, defined as ⊙(x,y) = xy + 1

It is commutative since: x ⊙ y = xy + 1 <=> yx + 1 = y ⊙ x

but not associative (whenever x ≠ z) since: x ⊙ (y ⊙ z) = x(yz + 1) + 1 = xyz + x + 1 ≠ (x ⊙ y) ⊙ z = (xy + 1)z + 1 = xyz + z + 1

Commutativity and Symmetry

Commutativity of binary operations corresponds to the symmetry of binary relations. A binary relation is said to be symmetric if the relation applies regardless of the order of its operands. For example, equality is symmetric as two equal mathematical objects are equal regardless of their order.

binary operations : commutativity ≋ binary relations : symmetry