Math Problem Statement

Write B ∪ C and B ∩ C using interval notation. The sets are defined as follows: B = {x | x ≥ 4} and C = {x | x > 9}.

Solution

The sets BB and CC are defined as:

B={xx4}B = \{ x \mid x \geq 4 \} C={xx>9}C = \{ x \mid x > 9 \}

We are tasked with finding BCB \cup C (union of BB and CC) and BCB \cap C (intersection of BB and CC) using interval notation.

1. BCB \cup C (Union):

The union of two sets includes all elements that are in either set. Since B={xx4}B = \{ x \mid x \geq 4 \} and C={xx>9}C = \{ x \mid x > 9 \}, the union will cover all values starting from x=4x = 4 (the lower bound of BB) and going to infinity. Hence, the union is:

BC=[4,)B \cup C = [4, \infty)

2. BCB \cap C (Intersection):

The intersection of two sets includes only the elements that are common to both sets. Since BB includes all values greater than or equal to 4, and CC includes all values greater than 9, their intersection will be all values greater than 9. Hence, the intersection is:

BC=(9,)B \cap C = (9, \infty)

Final Answer:

  • BC=[4,)B \cup C = [4, \infty)
  • BC=(9,)B \cap C = (9, \infty)

Would you like more details or further clarifications?

Here are five follow-up questions:

  1. What is the difference between open and closed intervals?
  2. How do we determine the union and intersection when sets are defined with inequalities?
  3. What happens if two sets have no common elements?
  4. How do unions and intersections apply to more than two sets?
  5. How do we handle cases where the sets are defined on finite intervals?

Tip: In set theory, always pay attention to whether the boundaries are included (closed intervals) or not (open intervals), as this changes the result significantly.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Union of Sets
Intersection of Sets
Interval Notation

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12