Math Problem Statement

Let U = set of all real numbers, P ={x|-5< or equal to x < 10}, Q={x|4<x<20}, and R={x|x, or equal to 6}: write in interval notation: PUQ

Solution

Let's break this down:

  • P={x5x<10}P = \{ x \mid -5 \leq x < 10 \} is the set of real numbers from 5-5 to 1010, including 5-5 but excluding 1010.

    • In interval notation: [5,10)[-5, 10).
  • Q={x4<x<20}Q = \{ x \mid 4 < x < 20 \} is the set of real numbers from 44 to 2020, excluding both endpoints.

    • In interval notation: (4,20)(4, 20).

Now, we want to find PQP \cup Q, the union of sets PP and QQ.

  • The union of two sets includes all elements that are in either set.
  • Looking at the intervals:
    • P=[5,10)P = [-5, 10)
    • Q=(4,20)Q = (4, 20)

When combining these intervals, we consider the overlap and non-overlap:

  • From 5-5 to 1010 is covered by PP, and from 44 to 2020 is covered by QQ.
  • The interval from 44 to 1010 is shared, but we need to combine all elements from 5-5 to 2020 (with the exception of the gap at 1010, which is excluded from PP but included in QQ).

Thus, PQP \cup Q is:

[5,20)[-5, 20)

This includes all real numbers from 5-5 to 2020, with 5-5 included and 2020 excluded.


Would you like more details or have any other questions?

Here are 5 related questions you might find helpful:

  1. What does the intersection of two sets mean, and how would we calculate it for PP and QQ?
  2. How does the union of intervals work in real-world scenarios?
  3. Can the union of two sets have gaps, or do they always form continuous intervals?
  4. How does the concept of a set relate to the number line in mathematics?
  5. What is the difference between inclusive and exclusive boundaries in interval notation?

Tip: Always pay attention to whether the endpoints are included or excluded (denoted by square brackets [][] or parentheses ()()) when working with intervals!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Set Theory
Interval Notation
Union of Sets

Formulas

P = [-5, 10)
Q = (4, 20)
P ∪ Q = [-5, 20)

Theorems

Union of Sets
Interval Union

Suitable Grade Level

Grades 9-11