1. What Is the Cartesian Product?
The Cartesian product of two sets forms a new set made of ordered pairs. Each ordered pair takes its first element from the first set and its second element from the second set.
This operation helps link elements of two sets in all possible ways.
2. Notation
The Cartesian product of sets A and B is written as:
\( A \times B \)
This is read as “A cross B”.
3. Meaning of A × B
The set \( A \times B \) contains ordered pairs:
- first element from A
- second element from B
Every combination of one element from A and one from B appears once.
3.1. Symbolic Form
\( A \times B = \{ (a, b) \mid a \in A,\; b \in B \} \)
4. Examples
Here are some simple examples to understand the Cartesian product clearly:
4.1. Example 1
If:
A = \{1,2\}, \quad B = \{3,4\}
Then:
A \times B = \{(1,3), (1,4), (2,3), (2,4)\}
4.2. Example 2
A = \{a,b\}, \quad B = \{1\}
Then:
A \times B = \{(a,1), (b,1)\}
4.3. Example 3 (Different Order)
Note that:
A \times B \neq B \times A
because ordered pairs change position. For example:
B \times A = \{(3,1),(3,2),(4,1),(4,2)\}
5. Cartesian Product With One Set Having One Element
If one set has a single element, the product simply attaches that element to each element of the other set.
5.1. Example
\{x\} \times \{1,2,3\} = \{(x,1), (x,2), (x,3)\}
6. Number of Elements in A × B
If A has m elements and B has n elements, then:
|A \times B| = m \times n
because every element of A pairs with every element of B.
7. Use of Cartesian Product in Relations
The Cartesian product is the base for defining relations. A relation from A to B is any subset of \( A \times B \). This means relations are built from ordered pairs formed by the Cartesian product.
8. Use in Coordinate Geometry
The entire coordinate plane is based on the Cartesian product. For example:
\( \mathbb{R} \times \mathbb{R} \)
represents all possible ordered pairs of real numbers — which form all points in the plane.
9. Important Points
A few key ideas about the Cartesian product:
9.1. Key Ideas
- Order matters in ordered pairs.
- \( A \times B \) is not the same as \( B \times A \).
- Every element of A pairs with every element of B.
- Useful in defining relations and coordinate systems.