Problem 54 - Algebra for College Students

Question

Express each set in simplest interval form. (Hint: Graph each set and look for the intersection or union.) $$ [-1,2] \cup(0,5) $$

Step-by-Step Solution

Answer

[ - 1 , 5 )

Step 1 - Graph the Sets on a Number Line

Graph the intervals - The interval [-1, 2] includes all numbers from -1 to 2, inclusive. - The interval (0, 5) includes all numbers from 0 to 5, exclusive.

Step 2 - Identify the Union of the Sets

The union ([-1, 2] nion ( 2]) includes all values that are in either set nion include the intervalseither.

Step 3 - Write the Simplified Form

Combine the intervals [-1,2] and (0, 5)Taking the unionoverlap. Therefore,the and [0,5) to form the single interval-1, 5).

Key Concepts

Union of Sets

To understand union of sets, think of it as combining all the elements from both sets. When we take the union of two sets, we include any element that appears in either set or both. In mathematical terms, the union of sets A and B, written as \(A \cup B\), includes every element belonging to A, or to B, or to both.
For example, let's look at the intervals in the exercise:

• Set A: \([-1, 2]\)
• Set B: \((0, 5)\)

The union \([-1, 2] \cup (0, 5)\) includes all numbers from -1 to 5, but we need to be careful about which ends are inclusive or exclusive:

• From -1 to 0, numbers are only in set A, so the interval is \([-1,0)\)
• From 0 to 2, numbers are in both sets, but since set B excludes 0, we write \((0,2]\)
• From 2 to 5, numbers are only in set B, so the interval is \((2, 5)\)

Thus, the union of the intervals is \([-1, 5)\).

Graphing Intervals

Graphing intervals on a number line is a helpful visual technique. It allows us to see which numbers are included in a set and where sets overlap.
Let's break it down for the given example.

• Interval \([-1, 2]\) includes all numbers from -1 to 2 and is shown with a solid line from -1 to 2, including the endpoints with solid dots.
• Interval \((0, 5)\) includes numbers from just after 0 to just before 5, shown with a solid line between 0 and 5, with open dots at 0 and 5 since those points are not included.

By combining these intervals on the same number line, we see that they overlap from 0 to 2. Solid dots and solid lines indicate included points (closed interval), while open dots indicate points that are not included (open interval). This visual makes it easy to spot which intervals to use when writing the union.

Set Notation

Set notation helps us describe intervals and sets clearly. It uses symbols like brackets and parentheses to show which endpoints are inclusive or exclusive.

• \((a, b)\) means all numbers between a and b, but *not* including a and b. This is called an open interval.
• \([a, b]\) means all numbers from a to b, *including* a and b, known as a closed interval.
• Combining these, \((a, b]\) or \([a, b)\) are half-open (or half-closed) intervals where one endpoint is included, and the other is not.

When writing unions or intersections, use \(\cup\) for union and \(\cap\) for intersection.
For instance, in our exercise, the sets \([-1, 2] \cup (0, 5)\) combine the closed interval \([-1, 2]\) and the open interval \((0, 5)\). Simplified, we get the interval \([-1, 5)\).