Ordered pair

An ordered pair is a pair of two objects, usually denoted $(x, y)$, in which we consider the order of the two objects to be important. Thus, the ordered pair $(2, 3)$ is different from the ordered pair $(3, 2)$. This should be contrasted with the notion of set (or multiset), in which we have $\{2, 3\} = \{3, 2\}$. In general, we say two ordered pairs, $(x, y)$ and $(a, b)$ are the same if and only if $x = a$ and $y = b$.

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 $a>b$ at all times. In order to efficiently test possibilities, we should order $b$ after $a$ (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.