Editing Cartesian Product of Two Sets

= Cartesian Product of Two Sets - Definition and Step-by-Step Examples =

The [[Cartesian Product]] of two sets is the set of all possible '''ordered pairs''' where the first element comes from the first set and the second element comes from the second set.

== Definition ==

If <math>A</math> and <math>B</math> are two sets, then the [[Cartesian Product]] of <math>A</math> and <math>B</math>, denoted by <math>A \times B</math>, is defined as:

<math>
A \times B = \{ (a, b) \mid a \in A,\, b \in B \}
</math>

Each element of <math>A \times B</math> is an ordered pair <math>(a, b)</math>, where:
- <math>a</math> belongs to set <math>A</math>
- <math>b</math> belongs to set <math>B</math>

== Important Notes ==

* The order of sets matters: <math>A \times B</math> is generally not equal to <math>B \times A</math>.  
* If one of the sets is empty, the [[Cartesian Product]] is also empty.

== Step-by-Step Example 1 ==

Let  
<math>A = \{1, 2\}</math>  
<math>B = \{x, y\}</math>

Step 1: Identify all elements of <math>A</math> and <math>B</math>.  
- <math>A</math> has elements 1 and 2  
- <math>B</math> has elements <math>x</math> and <math>y</math>

Step 2: Form ordered pairs by taking each element from <math>A</math> and pairing it with each element from <math>B</math>:

<math>
A \times B = \{ (1, x), (1, y), (2, x), (2, y) \}
</math>

== Step-by-Step Example 2 ==

Let  
<math>C = \{a, b\}</math>  
<math>D = \{1, 2, 3\}</math>

Step 1: Elements of <math>C</math>: <math>a</math>, <math>b</math>  
Step 2: Elements of <math>D</math>: 1, 2, 3

Step 3: Make all ordered pairs:

<math>
C \times D = \{ (a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3) \}
</math>

== Example 3: [[Cartesian Product]] with Itself ==

Let  
<math>E = \{0, 1\}</math>

Then  
<math>
E \times E = \{ (0, 0), (0, 1), (1, 0), (1, 1) \}
</math>

This is useful in representing points in a 2D grid.

== Visual Meaning ==

In coordinate geometry, <math>A \times B</math> gives all possible coordinate points where the x-coordinate comes from set <math>A</math> and the y-coordinate from set <math>B</math>.

== Summary ==

* The [[Cartesian Product]] combines elements from two sets into ordered pairs.  
* It's used in coordinate geometry, databases, and relation mappings.  
* Always pay attention to the order of sets.

[[Category:Set Theory]]
[[Category:Mathematics]]
[[Category:Ordered Pairs]]