---
title: "Cartesian products as regions"
description: "Visualize Cartesian products as grids, strips, and filled regions, then connect ordered pairs to geometric constraints."
canonical_url: "https://fanout.sh/ml-math/lessons/sets-cartesian-products-as-regions-t-09634sc"
md_url: "https://fanout.sh/ml-math/lessons/sets-cartesian-products-as-regions-t-09634sc.md"
access: "public"
---

# Cartesian products as regions

Visualize Cartesian products as grids, strips, and filled regions, then connect ordered pairs to geometric constraints.

## Public overview

Sketch Cartesian products as point grids, strips, or filled regions by analyzing each coordinate set and its boundary status.

Cartesian Products as Coordinate Spaces

For sets \(A\) and \(B\), the Cartesian product records all ordered pairs whose first coordinate comes from \(A\) and whose second coordinate comes from \(B\): \[A\times B=\{(a,b):a\in A,\ b\in B\}.\]

The most familiar visual example is \(\mathbb R^2\). Since \[\mathbb R^2=\mathbb R\times\mathbb R=\{(x,y):x,y\in\mathbb R\},\] every real \(x\)-coordinate can pair with every real \(y\)-coordinate.

Geometrically, that means \(\mathbb R^2\) is the entire Cartesian plane. A sketch of \(\mathbb R^2\) shades the whole plane, not just a curve, a grid, or selected points.

The same idea extends to three dimensions: \[\mathbb R^3=\mathbb R\times\mathbb R\times\mathbb R=\{(x,y,z):x,y,z\in\mathbb R\}.\] Its elements are all triples of real numbers, so \(\mathbb R^3\) describes every point in three-dimensional Cartesian space.

Sketching a Cartesian Product with Discrete x-Values

Consider \(S=\{1,2\}\times(0,1)\). By the Cartesian-product definition, \[S=\{(x,y):x\in\{1,2\},\ y\in(0,1)\}.\] The \(x\)-coordinate must be either \(1\) or \(2\), while the \(y\)-coordinate can be any real number strictly between \(0\) and \(1\).

Split the product into two pieces. The first piece fixes \(x=1\): \[S 1=\{(1,y):y\in(0,1)\}.\] In the coordinate plane, this is a vertical segment at \(x=1\), running between \(y=0\) and \(y=1\), with both endpoints excluded.

The second piece fixes \(x=2\): \[S 2=\{(2,y):y\in(0,1)\}.\] This gives a second vertical open segment, parallel to the first.

The full set is the union \(S=S 1\cup S 2\). A correct sketch shows exactly two vertical open line segments, one from \((1,0)\) to \((1,1)\) and one from \((2,0)\) to \((2,1)\), with open circles at all four endpoints because \((0,1)\) excludes both \(0\) and \(1\).

Visualizing Countably Many Vertical Rays

Now consider \[\mathbb N\times\mathbb R^-=\{(x,y):x\in\mathbb N,\ y\in\mathbb R^-\}.\] The first coordinate is restricted to the discrete positions \(1,2,3,\ldots\), while the second coordinate can be any negative real number.

The first slice fixes \(x=1\): \[S 1=\{(1,y):y\in\mathbb R^-\}\subset\mathbb N\times\mathbb R^-.\] This is a vertical ray below \(x=1\), extending downward from the \(x\)-axis, with \(y=0\) excluded.

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