Cartesian Product¶

Definition¶

The Cartesian product of two sets $A$ and $B$ is the set of all ordered pairs whose first component comes from $A$ and whose second component comes from $B$:

Interpretation¶

The Cartesian product $A\times B$ collects every possible pairing of an element of $A$ with an element of $B$, recorded as an ordered pair with the element of $A$ first.

With $A=\{a,b,c\}$ along the top and $B=\{1,2\}$ along the side, each cell of the grid holds one pair: one column for each element of $A$ and one row for each element of $B$, so $A\times B$ has $3\cdot2=6$ elements.

Order matters in two ways. Within a pair, $(a,1)$ and $(1,a)$ are different objects. Between the factors, $A\times B$ and $B\times A$ are different sets unless $A=B$ or one of the sets is empty; here $(a,1)\in A\times B$ but $(a,1)\notin B\times A$$A$.

The Cartesian coordinate system is the case $\mathbb{R}\times\mathbb{R}=\mathbb{R}^2$${\mathbb{R}}^2$, where every point of the plane is a pair $(x,y)$ of real numbers, and the sample space of two dice is $\{1,\ldots,6\}\times\{1,\ldots,6\}$$\{1,\dots ,6\}$.

Key Results¶

• If $A$ and $B$ are finite, then $\operatorname{card}(A\times B)=\operatorname{card}(A)\operatorname{card}(B)$$\mathrm{card}(A)\mathrm{card}(B)$.
• Cartesian products are used to define binary operations, since a binary operation on $A$ is a function from $A\times A$ to $A$.

Examples¶

• If $A=\{1,2\}$ and $B=\{x,y\}$$\{x,y\}$, then $A\times B=\{(1,x),(1,y),(2,x),(2,y)\}$$\{(1,x),(1,y),(2,x),(2,y)\}$.
• The set $\mathbb{R}^2$ can be understood as $\mathbb{R}\times\mathbb{R}$$\mathbb{R}$, the set of all ordered pairs of real numbers.

Prerequisites¶

• Set

Related Concepts¶

• Function
• Euclidean Space
• Sample Space

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: product set, product of sets Keywords: direct product Symbols: element of, real numbers

Last updated: 20 October 2026 03:02