Question

Sketch these Cartesian products on the \(x-y\) plane \(\mathbb{R}^{2}\) (or \(\mathbb{R}^{3}\) for the last two). $$ [-1,1] \times[1,2] $$

Short answer

The Cartesian product \([-1,1] \times[1,2]\) represents a rectangle on the \(x-y\) plane with corners at (-1, 1), (-1, 2), (1, 1) and (1, 2).

Step-by-step solution

Understand the intervals

The first interval \([-1, 1]\) is on the x-axis and it means that x can take any value from -1 to 1 inclusive. Likewise, the second interval \([1, 2]\) represents the y-axis and it means y can take any value from 1 to 2 inclusive.

Identify the Cartesian product

The Cartesian product of two sets is defined as the set of all ordered pairs in which the first component is from the first set and the second component is from the second set. In this case, the Cartesian product of the two intervals is \{ (x, y) where -1 ≤ x ≤ 1 and 1 ≤ y ≤ 2 \}.

Sketch the set of ordered pairs on the \(x-y\) plane

This Cartesian product represents a rectangle on the \(x-y\) plane, with corners at (-1, 1), (-1, 2), (1, 1) and (1, 2). The length of the rectangle extends along the x-axis from -1 to 1, and the width extends along the y-axis from 1 to 2.

Key concepts

Interval Notation

Understanding interval notation is crucial when dealing with mathematical concepts such as functions, sequences, and in this case, Cartesian products. Interval notation is a shorthand way to describe a range of values, typically on a number line.

In an interval, the numbers at the ends are called the bounds and these can be included in the interval or not. Inclusive bounds are denoted using square brackets \[ \], meaning that the values at the ends are part of the interval. For instance, the interval \[ -1, 1 \] includes all numbers from -1 to 1, including the endpoints. Exclusive bounds are denoted with parentheses \( \) and exclude the endpoints. For example, \( -1, 1 \) includes numbers between -1 and 1, but not -1 and 1 themselves.

When we look at the cartesian product, such as \[ -1, 1 \] \times \[ 1, 2 \], interval notation helps us understand the specific set of x and y values that will be included in our set of ordered pairs. To improve comprehension, one can visualize these intervals on a number line before considering their Cartesian product to better grasp the range of values each variable can take.

Ordered Pairs

An ordered pair is a fundamental component in coordinate systems, representing the position of a point in a two-dimensional context. Each pair is written in the form \(x, y\), where \(x\) represents the position on the horizontal axis and \(y\) represents the position on the vertical axis.

To understand the concept of ordered pairs in relation to Cartesian products, think of two separate number lines intersecting at the point \(0,0\). For any point on the plane, you pick a value from one number line for the x-coordinate and one from the second number line for the y-coordinate. In the context of the exercise, we take an x value from the interval \[ -1, 1 \] and a y value from \[ 1, 2 \] to form our ordered pairs. When dealing with a Cartesian product, every possible combination of \(x\) and \(y\) from the specified intervals is taken, which illustrates a key principle of Cartesian products: the pairing of every element of one set with every element of another.

Sketching on Coordinate Plane

Sketching the Cartesian product on a coordinate plane involves visualizing the set of all ordered pairs. Begin by drawing the two perpendicular number lines, typically referred to as the x-axis and y-axis. Identify and mark the intervals given, in this exercise \[ -1, 1 \] on the x-axis and \[ 1, 2 \] on the y-axis.

Next, visualize the rectangle formed by these intervals. Draw horizontal and vertical lines emanating from the interval endpoints, creating a rectangular area that represents all the points that fall within the specified ranges. The Cartesian product \[ -1, 1 \] \times \[ 1, 2 \] is represented as the rectangle bounded by these lines, with vertices at the points \( -1, 1 \), \( -1, 2 \), \( 1, 1 \), and \( 1, 2 \). Every point inside this rectangle, including the edges and vertices, is a part of the Cartesian product. This visual representation on the coordinate plane greatly aids in understanding how two intervals interact with each other when taking their Cartesian product.