A Cartesian product of two sets is a new set that is constructed from the two sets. In order to define Cartesian products, we need to define a mathematical object called an ordered pair.
An ordered pair is an ordered list of two mathematical objects, and , written as . The objects in an ordered pair are called components. The object is the first component of , and the object is the second component of .
To form the Cartesian product , we pair each element of , placed in the first component of the ordered pair, with each element of , placed in the second component of the ordered pair.
The set contains all ordered pairs whose first entry is an element of the set and whose second entry is an element of the set . We write ordered pairs whose first entry is and whose second entry is as . We get
So, two ordered pairs are equal if they have matching first components and matching second components. The fact that the elements of are called ordered pairs indicates that we must pay attention to order for Cartesian products. In comparison, recall that the order of the elements in a set given in roster form does not matter. (See Example 5.20.)
Since the empty set does not contain any elements, there are no elements to be placed into the second component of the Cartesian product . So, we have that for any set . Similarly, for any set .