Planar regions in set-builder form

Encode planar regions with ordered pairs, inequalities, and logical connectors while preserving every included and excluded boundary.

Describing a Rectangle by Its Boundaries

A region in the coordinate plane is a set of ordered pairs, so its set-builder description should name both the ambient space and the coordinate restrictions. Here the ambient set is \(\mathbb R^2\), meaning each point has the form \((x,y)\) with real-valued coordinates.

For this worked rectangle, the horizontal span runs from \(x=2\) to \(x=6\), and the vertical span runs from \(y=2\) to \(y=5\). If only the interior were included, the coordinate restrictions would be \(2<x<6\) and \(2<y<5\).

The boundary markings decide which strict inequalities become non-strict. The left and right sides are included, so \(x=2\) and \(x=6\) must both be allowed; this gives \(2\le x\le 6\). The top and bottom edges are excluded, so the vertical restriction stays strict: \(2<y<5\).

Putting the coordinate domain and both restrictions together gives \[\{(x,y)\in\mathbb R^2:2\le x\le 6,\ 2<y<5\}.\] The comma between the restrictions functions as an and: a point belongs to the region only when its \(x\)-coordinate and its \(y\)-coordinate pass their tests.

Describing a Finite Rectangular Lattice Set

Not every rectangular picture represents every real point inside a rectangle. If the marked points are lattice points, the ambient set should be \(\mathbb Z^2\), not \(\mathbb R^2\). Writing \((x,y)\in\mathbb Z^2\) says both coordinates must be integers.

The horizontal coordinates in this example are \(1,2,3\), summarized by \(1\le x\le 3\). Both inequalities are non-strict because the left boundary \(x=1\) and the right boundary \(x=3\) are included as columns of points.

The vertical coordinates are \(-3,-2,-1,0,1,2\), summarized by \(-3\le y\le 2\). Again, both inequalities are non-strict because the bottom row \(y=-3\) and the top row \(y=2\) are included.

Combining the integer-coordinate domain with both coordinate restrictions gives \[\{(x,y)\in\mathbb Z^2:1\le x\le 3,\ -3\le y\le 2\}.\] A point belongs only if its \(x\)-coordinate is one of the allowed columns and its \(y\)-coordinate is one of the allowed rows.

Integer Lattice Points on or Below a Line

This region is not a continuous half-plane in \(\mathbb R^2\); it consists of plotted lattice points. Each selected point has integer coordinates, so the ambient domain is \(\mathbb Z^2\), equivalently \(\mathbb Z\times\mathbb Z\).

The boundary is the line \(y=x+2\). Points below that line satisfy the strict inequality \(y<x+2\), since their \(y\)-coordinate is smaller than the line's height at the same \(x\)-value.