Cartesian Product of Sets

1. What the Cartesian Product Means

The Cartesian product of two sets forms a new set made up of ordered pairs. Each pair takes its first element from the first set and its second element from the second set. This gives all possible combinations of two sets.

2. Notation

The Cartesian product of A and B is written as:

\( A \times B \)

and is read as “A cross B”.

2.1. Symbolic Definition

\( A \times B = \{(a,b) \mid a \in A,\ b \in B\} \)

3. How It Forms Ordered Pairs

Every element of A pairs with every element of B. No combination is skipped, and no pair repeats. The order matters because the pair \( (a,b) \) is different from \( (b,a) \) unless both elements are the same.

3.1. Example

Let:

A = \{1,2\}, \quad B = \{3,4\}

Then:

A \times B = \{(1,3),(1,4),(2,3),(2,4)\}

4. Size of the Cartesian Product

If A has m elements and B has n elements, then the product has m × n ordered pairs.

4.1. Formula

|A \times B| = |A| \cdot |B|

4.2. Example

If |A| = 3 and |B| = 2, then:

|A \times B| = 3 \times 2 = 6

5. Cartesian Product With One Set Having One Element

If one of the sets has only a single element, then each pair takes that element as its fixed part.

5.1. Example

\{x\} \times \{1,2,3\} = \{(x,1),(x,2),(x,3)\}

6. Order Matters in a Cartesian Product

The pair (a,b) is not the same as (b,a) when a ≠ b. So usually:

A \times B \neq B \times A

6.1. Example

Using the earlier sets:

B \times A = \{(3,1),(3,2),(4,1),(4,2)\}

This is different from A × B.

7. Cartesian Product in Relations

The Cartesian product is the base for defining relations because a relation is any subset of \( A \times B \). Once all ordered pairs are listed, choosing certain pairs creates a relation.

7.1. Example

A = \{1,2\},\ B = \{3,4\}