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.