Cartesian Product of Sets
Ordered Pairs
In sets, the order of elements is not important. For example, the sets {2, 3} and {3, 2} are equal to each other. However, there are many instances in mathematics where the order of elements is essential. So, for example, the pairs of numbers with coordinates (2, 3) and (3, 2) represent different points on the plane. This leads to the concept of ordered pairs.
An ordered pair is defined as a set of two objects together with an order associated with them. Ordered pairs are usually written in parentheses (as opposed to curly braces, which are used for writing sets).
In the ordered pair
Two ordered pairs
The equality
Tuples
The concept of ordered pair can be extended to more than two elements. An ordered
Similarly to ordered pairs, the order in which elements appear in a tuple is important. Two tuples of the same length
Unlike sets, tuples may contain a certain element more than once:
Ordered pairs are sometimes referred as
Cartesian Product of Two Sets
Suppose that
The Cartesian product is also known as the cross product.
The figure below shows the Cartesian product of the sets
It consists of
Similarly, we can find the Cartesian product
As you can see from this example, the Cartesian products
If
Cartesian Product of Several Sets
Cartesian products may also be defined on more than two sets.
Let
If
Some Properties of Cartesian Product
- The Cartesian product is non-commutative:
if only if either or- The Cartesian product is non-associative:
- Distributive property over set intersection:
- Distributive property over set union:
- Distributive property over set difference:
- If
then for any set
Cardinality of Cartesian Product
The сardinality of a Cartesian product of two sets is equal to the product of the cardinalities of the sets:
Similarly,