Problem 38 - Calculus: One and Several Variables

Question

Sketch the set on a number line. \(\left[-3, \frac{3}{2}\right) \cap\left(\frac{3}{2}, \frac{5}{2}\right]\).

Step-by-Step Solution

Answer

The intersection of the intervals is an empty set, represented as \(\emptyset\) or {}. On the number line, no points are marked.

Step 1 - Understand the given intervals

The first interval includes all numbers from -3 to \(\frac{3}{2}\), not including \(\frac{3}{2}\). This is represented as \([-3, \frac{3}{2})\). The second interval includes all numbers\tgreater than \(\frac{3}{2}\) and less than or equal to \(\frac{5}{2}\). This is represented as \((\frac{3}{2}, \frac{5}{2}]\).

Step 2 - Determine the intersection of the intervals

Note that the number \(\frac{3}{2}\) is not included in either of the intervals. Therefore, there are no numbers that are common to both intervals. Thus the intersection of these two intervals is an empty set. This can be represented as \(\emptyset\) or {}\.

Step 3 - Sketch on the Number Line

An empty set can be represented on the number line by simply not marking any points.

Key Concepts

Number Line

The number line is essential in visualizing the position and relationship of numbers. Think of it as a horizontal line on which every point corresponds to a specific number. This line stretches infinitely in both directions, with numbers getting larger as you move to the right and smaller as you move to the left. In the solution to the exercise, a number line would be used to represent the given intervals.
Intervals are visual segments or portions of the number line, and they can illustrate ranges of numbers such as 'all numbers between -3 and 3/2'. To sketch the first interval \( [-3, \frac{3}{2}) \) on a number line, one would draw a line from -3, including -3 (denoted by a filled-in circle or bracket), and up to but not including \(\frac{3}{2}\) (denoted by an open circle or parenthesis). However, because the intersection in the exercise results in an empty set, there would be no segment to show on the number line representing the intersection.
When teaching this concept, it's important to illustrate how different markers (like brackets or parentheses) are used to indicate whether an endpoint is included in the interval.

Interval Notation

Interval notation is a shorthand used to describe ranges of continuous numbers on the number line. When we talk about interval notation, we refer to a system that uses brackets and parentheses to represent intervals. A set of numbers between two endpoints is enclosed within these brackets or parentheses, with square brackets [ ] indicating that the endpoints are included, and parentheses ( ) indicating that they are not.
In the provided exercise, the interval \( [-3, \frac{3}{2}) \) means that -3 is included in the interval but \(\frac{3}{2}\) is not, hence the use of a square bracket at -3 and a parenthesis at \(\frac{3}{2}\). Similarly, the expression \( (\frac{3}{2}, \frac{5}{2}] \) includes numbers greater than but not including \(\frac{3}{2}\) and up to and including \(\frac{5}{2}\).
One key point to emphasize in teaching interval notation is the clarification of endpoint inclusion or exclusion, which directly affects the intersection of intervals in exercises like the one given.

Empty Set

An empty set, symbolized by \(\emptyset\) or {}, is a set with no elements. In the context of our number line and interval notation, it means there are no numbers that satisfy the condition being considered; in other words, it is an interval with no numbers in it.
In the exercise solution, the intersection between the two intervals is an empty set because they do not share any common numbers. The first interval stops just before \(\frac{3}{2}\), and the second interval starts immediately after. Since there is no overlap, no points on the number line would represent the intersection. The concept of an empty set is fundamental in set theory and is important to grasp as it affects understanding of operations like intersection and union in more complex mathematical domains.
Emphasizing the significance of an empty set in various mathematical scenarios can help students understand situations where no solutions are applicable or when two sets have no elements in common.