Question

Sketch the following sets of points in the \(x-y\) plane. $$ \\{(x, y): x \in[0,1], y \in[1,2]\\} $$

Short answer

The sketch is a square defined by the points (0,1), (0,2), (1,1), and (1,2) in the \(x-y\) plane. All points inside this square and on the edges are included in the set.

Step-by-step solution

Understand the Ranges

First, let's understand that for every \(x\) in the range \([0,1]\), there is a \(y\) in the range \([1,2]\). This translates to all the points \( (x,y) \) where \(x\) varies between 0 and 1 (inclusive) and \(y\) varies between 1 and 2 (inclusive).

Identify the Points at the-extremes

Consider the starting and ending points of the ranges. These points are (0,1), (0,2), (1,1), and (1,2). These will serve as the markers for our plot.

Draw the Graph

Now, for the actual sketching part. Draw a square in the plane enclosed by these points. All points inside this square and on the edges are included in the defined set.

Key concepts

Cartesian Plane

The Cartesian plane is a fundamental concept in coordinate geometry. It is a two-dimensional surface on which we can plot points, lines, and curves. The plane is defined by two perpendicular number lines that intersect at what is called the origin, which has the coordinates \(0, 0\).

The horizontal number line is known as the x-axis, and the vertical number line is called the y-axis. Each point on the Cartesian plane is described by a pair of numerical coordinates: \(x, y\), where \(x\) represents the horizontal position, and \(y\) represents the vertical position.

To graph a set of points on the Cartesian plane, you start at the origin, move along the x-axis by the amount specified by the \(x\) coordinate and then move parallel to the y-axis by the \(y\) coordinate. For example, the coordinate \(1, 2\) means a point is located 1 unit to the right of the origin and 2 units up from the x-axis.

Set Notation

Set notation is a system of symbols used to specify collections of objects or numbers. In mathematics, sets are typically enclosed in curly braces \(\{ \{\}\), and the elements within the set are separated by commas. When dealing with coordinate points, the \(x\) and \(y\) values within the point \( (x, y) \) are not separated by a comma but are enclosed within parentheses to indicate that they are an ordered pair.

In the context of graphing, set notation allows us to express the range of allowable values for the x and y coordinates. For example, \( x \in [0,1]\) means \(x\) can take any value between 0 and 1, including the end points. The notation \(y \in [1,2]\) specifies that \(y\) can be any number from 1 to 2, inclusive. The colon \( : \) following the set bracket is read as 'such that', indicating a condition for the sets elements.

Graphing Ranges

Graphing ranges involve plotting all the possible values that fall within a given interval on a graph. Ranges can be continuous, including every number between two endpoints, or they can be discrete, including only certain defined numbers. When the ranges involve inequalities, we often denote whether the endpoints are included or excluded using square brackets \( [ ] \) for inclusion, and parentheses \( ( ) \) for exclusion.

In the original exercise, the ranges for \(x\) and \(y\) are continuous and both include their endpoints, designated by the use of square brackets. These ranges \(x \in [0,1]\) and \(y \in [1,2]\) create a rectangle or square shape on the Cartesian plane when graphed. To visualize the range, you would draw a line for every value of \(x\) from 0 to 1, and for every line of \(x\), you would draw a point for every value of \(y\) from 1 to 2, filling in the square completely.