2.1 Cross Product of Sets

Types: N/A
Examples: 2.2 Examples of Cartesian Products
Constructions: N/A
Generalizations: 1.1 Set,

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Cross Product of Sets AKA Cartesian Product

Let $A$ and $B$ be sets. The cross product of $A$ and $B$ is the set

A \times B = {(a, b) : a \in A and b \in B} .

The notation $A \times B$ is read " $A$ cross $B$ ."

The cross product $A \times B$ is the set of all ordered pairs that can be made by first choosing coordinates from the set $A$ and second coordinates from $B$ . The cross product is also known as the Cartesian Product.

Warning

The cross product between to sets is different from the cross product between vectors.

Elements of the cross products of sets take the form $(a, b)$ , known as an ordered-pair formed from elements $a \in A$ and $b \in B$ .

first coordinate: $a$
second coordinate: $b$

We say two ordered pairs $(a, b)$ and $(x, y)$ are equal iff $a = x$ and $b = y$ . Changing either coordinate of a given ordered pair $(a, b)$ creates a different ordered pair.