# Ordered pair

An **ordered pair** is a pair of two objects, usually denoted , in which we consider the order of the two objects to be important. Thus, the ordered pair is different from the ordered pair . This should be contrasted with the notion of set (or multiset), in which we have . In general, we say two ordered pairs, and are the same if and only if and .

The notion of an ordered pair can be naturally extended to that of an ordered tuple.

Order is necessary, when things aren't commutative. Also assume we have a restriction in a problem, such that at all times. In order to efficiently test possibilities, we should order after (to input its value into calculating the minimum b) in any programming or math. We don't waste time, to figure out already known impossible solutions, in this implementation.

## Formal Definition

In the language of set theory, it is not trivial to define an ordered pair since the set and are equivalent. Thus, the definition of an ordered pair is the set Through this definition, the pair does not equal the pair since the set and are not equivalent. However, for the ordered pair the resulting set reduces to (do you see why?). Thus reversing the positions of in the ordered pair does not change the resulting set.