Math Problem Statement

for each of the following sets decide whether it is an interval .if it is ,write it in interval notation. if it is not justify your decision 1. [-2,5)union(3,7), 2. (-infinity,0)union(0,infinity) 3. [-2,2)union[2,5] 4. (-infinity,3]union[5,8)

Solution

Let's go through each set and analyze whether it forms an interval. An interval is a set of real numbers that contains all numbers between any two numbers in the set. Intervals can be open, closed, or half-open.

1. [2,5)(3,7)[-2, 5) \cup (3, 7)

  • The set consists of two intervals: [2,5)[-2, 5) and (3,7)(3, 7).
  • These intervals do not overlap. In fact, there is a gap between 5 and 3, meaning the union does not form a continuous set of numbers.

Thus, this is not an interval.

Reason: There is a break between 5 and 3.

2. (,0)(0,)(- \infty, 0) \cup (0, \infty)

  • This set consists of two intervals: (,0)(- \infty, 0) and (0,)(0, \infty).
  • These intervals do not overlap either, and there is a gap at 0 where the set does not include the number 0.

Thus, this is not an interval.

Reason: There is a gap at 0.

3. [2,2)[2,5][-2, 2) \cup [2, 5]

  • This set consists of two intervals: [2,2)[-2, 2) and [2,5][2, 5].
  • These intervals do overlap at the point 2, since the number 2 is included in both intervals.
  • The union forms a continuous set from 2-2 to 5.

Thus, this is an interval, and it can be written as:

[2,5][-2, 5]

4. (,3][5,8)(- \infty, 3] \cup [5, 8)

  • This set consists of two intervals: (,3](- \infty, 3] and [5,8)[5, 8).
  • These intervals do not overlap, and there is a gap between 3 and 5.

Thus, this is not an interval.

Reason: There is a gap between 3 and 5.


Summary of answers:

  1. Not an interval (gap between 5 and 3).
  2. Not an interval (gap at 0).
  3. Interval: [2,5][-2, 5].
  4. Not an interval (gap between 3 and 5).

Would you like a more detailed explanation of any specific part? Or do you have other questions related to this?

Further questions:

  1. What makes two intervals overlap or not overlap?
  2. How do you determine the gap between intervals?
  3. Can you form intervals by combining more than two sets?
  4. How do open and closed intervals affect their properties?
  5. How do you handle infinity in interval notation?

Tip: When determining if two sets form an interval, always check if there are any gaps between the sets. If there is even a small gap, it's not a continuous interval.

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Math Problem Analysis

Mathematical Concepts

Intervals
Set Theory
Union of Sets
Real Numbers

Formulas

Interval notation: [a, b], (a, b), [a, b), (a, b)

Theorems

Properties of intervals
Union of intervals

Suitable Grade Level

Grades 9-12