Characteristic properties of tuples

The general rule for the identity of two -tuples is

if and only if

Thus a tuple has properties that distinguish it from a set.

1. A tuple may contain multiple instances of the same element, so
tuple ; but set = .

2. Tuple elements are ordered: tuple , but set .

3. A tuple has a finite number of elements, while a set or a multiset may have an infinite number of elements.

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematics started to become formalized. By contrast, division and subtraction are not commutative.

The term "commutative" is used in several related senses.^[7][8]

1. A binary operation on a set S is called commutative if:

An operation that does not satisfy the above property is called noncommutative.

2. One says that x commutes with y under if:

3. A binary function is called commutative if: