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Chapter 0: Problem 19

Write each union as a single interval. $$[2,7) \cup[5,20)$$

Short Answer

Expert verified
The union of the given intervals \([2,7)\) and \([5,20)\) can be written as a single interval: \([2, 20)\).

Step by step solution

01

Identify the intervals

The given exercise includes two intervals: Interval A: \([2,7)\) Interval B: \([5,20)\)
02

Determine the lowest endpoint

Compare the endpoints of both intervals: Interval A starts at 2, while interval B starts at 5. Choose the smallest value (2) as the starting point of the union.
03

Determine the highest endpoint

Compare the endpoints of both intervals: Interval A ends just before reaching 7, while interval B ends just before reaching 20. Choose the largest value (20) as the endpoint of the union.
04

Write the union as a single interval

With the endpoints determined in steps 2 and 3, the union as a single interval is: \([2, 20)\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Union of intervals
When working with intervals, the union operation is essential to understand. The union of intervals combines two or more intervals into a single set that contains all the numbers present in either interval. For example, if we have two intervals,
  • \([2,7)\)
  • \([5,20)\)
Joining these intervals involves merging them to encompass all the numbers from 2 up to, but not including, 20. The union is represented as \([2, 20)\), which covers all numbers found in either of the original intervals, minus any overlap repetition.
Understanding unions helps simplify complex sets, making problems easier to handle.
Interval notation
Interval notation is a convenient way of writing the set of all numbers between two endpoints. It is commonly used in mathematics to express intervals without listing all the numbers individually. When you write an interval, you indicate whether each endpoint is included or not using brackets:
  • "[ ]" means the endpoint is included (a closed interval).
  • "( )" means the endpoint is not included (an open interval).
If you have the interval \([2, 7)\), it means all real numbers starting from 2 and going up to, but not including, 7. Interval notation helps you quickly understand ranges and boundaries when dealing with sets of numbers.
Endpoints in intervals
Endpoints are crucial in defining intervals as they determine where an interval starts and ends. There are two types of endpoints:
  • Open endpoint, shown by a parenthesis, indicates the endpoint is not part of the interval.
  • Closed endpoint, indicated by a bracket, shows the endpoint is included.
In our exercise,
  • \([2, 7)\) includes the number 2 but not 7.
  • \([5, 20)\) includes the number 5 but not 20.
When forming a union, you combine the smallest and largest relevant endpoints to form a new interval. This helps ensure the interval accurately represents all numbers intended by the original sets.

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Most popular questions from this chapter

The intersection of two sets of numbers consists of all numbers that are in both sets. If \(A\) and \(B\) are sets, then their intersection is denoted by \(A \cap B .\) In Exercises \(47-\) \(56,\) write each intersection as a single interval. $$(3, \infty) \cap[2,8]$$

The intersection of two sets of numbers consists of all numbers that are in both sets. If \(A\) and \(B\) are sets, then their intersection is denoted by \(A \cap B .\) In Exercises \(47-\) \(56,\) write each intersection as a single interval. $$(-\infty,-10] \cap(-\infty,-8]$$

(a) Verify that $$\frac{16}{2}-\frac{25}{5}=\frac{16-25}{2-5}$$ (b) From the example above you may be tempted to think that $$\frac{a}{b}-\frac{c}{d}=\frac{a-c}{b-d}$$ provided none of the denominators equals \(0 .\) Give an example to show that this is not true.

Simplify the given expression as much as possible. $$\frac{x-3}{4}-\frac{5}{y+2}$$

Explain how you could show that \(51 \times 49=2499\) in your head by using the identity \((a+b)(a-b)=\) \(a^{2}-b^{2}\).
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