Question

Sketch the following sets of points in the \(x-y\) plane. $$ \\{(x, x+y): x \in \mathbb{R}, y \in \mathbb{Z}\\} $$

Short answer

The sketch of the set of points on the \(x-y\) plane forms a family of parallel lines, where each line corresponds to a different integer value for \(y\). The lines are separated by a vertical distance of 1 unit.

Step-by-step solution

Understand the Coordinates

The given set of points are of the form \((x, x+y)\). Here, \(x\) and \(y\) define the coordinates of the points. \(x\) can be any real number, so it ranges from \(-\infty\) to \(+\infty\). For the \(y\) coordinate, it's \(x+y\), where \(y\) can be any integer. This means that the points are translated vertically by an integer amount.

Analyzing the Effect of Changing \(x\)

As \(x\) changes, both the \(x\) coordinate and \(y\) coordinate of the points will change. Since \(x\) can be any real number, the points will form a continuous line for each individual value of \(y\).

Analyzing the Effect of Changing \(y\)

If we change the value of \(y\), the \(x\) coordinate remains unaltered while the \(y\) coordinate changes by the integer amount \(y\). Thus, an increase or decrease in \(y\) will result in the line being translated up or down in the plane.

Sketching the Set of Points

Drawing this on the \(x-y\) plane, it can be seen that for each integer \(y\), there is a continuous line. Increasing \(y\) by 1 shifts the line up by 1 unit. As such, the sketch is a family of parallel lines, separated vertically by 1 unit, extending to \(-\infty\) and \(+\infty\) in both axes.