Question

Sketch the set \(X=[-1,3] \times[0,2]\) on the plane \(\mathbb{R}^{2}\). On separate drawings, shade in the sets \(\bar{X}\) and \(\bar{X} \cap([-2,4] \times[-1,3])\)

Short answer

The set X is a rectangle located at the intervals [-1, 3] on the x-axis and [0, 2] on the y-axis. The set \(\bar{X}\) is the entire real plane except for this rectangle. The intersection \(\bar{X} \cap([-2,4] \times[-1,3])\) is the region outside the rectangle X but inside the larger rectangle defined by intervals [-2, 4] and [-1, 3].

Step-by-step solution

Sketch X

The set X is represented by the Cartesian product of the intervals [-1,3] and [0,2]. In order to sketch this set, plot two perpendicular lines representing the x-axis (ranging from -1 to 3) and the y-axis (ranging from 0 to 2). This rectangle will represent the set X.

Sketch \(\bar{X}\)

\(\bar{X}\) represents the complement of X. This is the area of the real plane that is not part of the set X. To sketch this, simply shade in everything outside the rectangle from Step 1.

Sketch \(\bar{X} \cap([-2,4] \times[-1,3])\)

This represents the intersection of the complement of X and the set represented by the Cartesian product [-2, 4] x [-1, 3]. This is the shared area between the shaded region from Step 2 and a new rectangle represented by the intervals [-2, 4] on the x-axis and [-1, 3] on the y-axis. To sketch this, start by drawing the new rectangle. Then, shade the region that is outside the original rectangle X, but inside this new larger rectangle.

Key concepts

Cartesian Product

The Cartesian product is a mathematical operation that returns a set from multiple sets. Each member of the resulting set is a pair consisting of one element from each of the input sets. For example, if we have two sets, Set A and Set B, the Cartesian product of these, denoted as A × B, pairs every element of Set A to every element of Set B.

When visualizing the Cartesian product on the coordinate plane, like in our exercise, it is akin to creating a rectangle that spans the range of both intervals on the respective axes. If we look at the exercise solution, the Cartesian product of the intervals \[[-1,3]\] and \[0,2\] forms a rectangle with corners at (-1,0), (-1,2), (3,0), and (3,2). This rectangle represents all possible pairs between the numbers within those intervals.

Understanding through Coordinates

Consider the point (2,1) – this point is within the bounds of our Cartesian product since 2 is between -1 and 3, and 1 is between 0 and 2, displaying how elements are paired in this visual context.

Complement of a Set

In set theory, the complement of a set, denoted by a bar (\(\bar{X}\)), includes everything that is not in the set within a given universe. It’s like drawing a circle around our set X on the plane and saying that the complement is everything outside this circle.

While working on our exercise, the complement of X would be every point on the plane that is not included within the rectangle defined by X. To sketch this, we shade everything but the rectangle from Step 1. This technique of shading helps to visually represent what is included in or excluded from the complement.

Complement in the Plane

Imagine the coordinate plane is an immense piece of paper. Our set X is a small window we have cut out. The complement is the rest of the infinite paper, excluding the window.

Interval Notation

Interval notation is a way of writing subsets of the real number line using intervals. It’s a compact form of expressing continuous sets of numbers within a certain range.

An interval expressed as \[a, b\] denotes all the numbers between a and b, including a and b themselves – this is known as a closed interval. On the other hand, \(a, b\) denotes all the numbers between a and b, not including a and b – an open interval. Our exercise uses closed intervals, such as \[[-1,3]\] and [0,2], which include the end points.

Interval Notation Tips

You can remember that square brackets, [ and ], like square corners, 'include' the end points, while the round parentheses, ( and ), 'exclude' the end points, just like their round shape suggests no corner (or end point) is included.

Intersection of Sets

The intersection of sets refers to the elements that two or more sets have in common. It is denoted by the symbol \(\cap\). When visualizing the intersection, it is the overlapping region of the sets involved.

In our exercise, the intersection is between the complement of set X and another rectangle on the plane represented by the Cartesian product \[[-2,4] \times [-1,3]\]. To sketch this, one may start with the second rectangle and then shade the portion which is also part of the first set's complement, effectively showing where the two sets overlap.

Finding Common Ground

If you're having trouble visualizing it, imagine two different colored pieces of transparent film, each with drawings on them. When laid over each other, the intersection is where the colors from both pieces mix; it's the shared area of the drawings.