Calculus

Set Theory

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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 the element is called the first entry or first component, and is called the second entry or second component of the pair.

Two ordered pairs and are equal if and only if and In general,

The equality is possible only if

Tuples

The concept of ordered pair can be extended to more than two elements. An ordered tuple is a set of objects together with an order associated with them. Tuples are usually denoted by The element is called the entry or component, and is called the length of the tuple.

Similarly to ordered pairs, the order in which elements appear in a tuple is important. Two tuples of the same length and are said to be equal if and only if for all So the following tuples are not equal to each other:

Unlike sets, tuples may contain a certain element more than once:

Ordered pairs are sometimes referred as tuples.

Cartesian Product of Two Sets

Suppose that and are non-empty sets. The Cartesian product of two sets and denoted is the set of all possible ordered pairs where and

The Cartesian product is also known as the cross product.

The figure below shows the Cartesian product of the sets and

Cartesian product of two sets A={1,2,3} and B={x,y}.
Figure 1.

It consists of ordered pairs:

Similarly, we can find the Cartesian product

As you can see from this example, the Cartesian products and do not contain exactly the same ordered pairs. So, in general,

If then is called the Cartesian square of the set and is denoted by

Cartesian Product of Several Sets

Cartesian products may also be defined on more than two sets.

Let be non-empty sets. The Cartesian product is defined as the set of all possible ordered tuples where and

If then is called the Cartesian power of the set and is denoted by

Some Properties of Cartesian Product

  1. The Cartesian product is non-commutative:
  2. if only
  3. if either or
  4. The Cartesian product is non-associative:
  5. Distributive property over set intersection:
  6. Distributive property over set union:
  7. Distributive property over set difference:
  8. 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,

See solved problems on Page 2.

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