https://file.aiquickdraw.com/mathbot/file/1730928813784esag44yc.jpg

Let's work through each part systematically to determine the elements of each set expression, given the universal set \( U = \{1, 2, 3, \dots, 10\} \) and the sets \( A = \{1, 2, 3, 4, 5\} \), \( B = \{1, 2, 4, 8\} \), \( C = \{1, 2, 3, 5, 7\} \), and \( D = \{2, 4, 6, 8\} \).

### Step-by-Step Solutions

#### (a) \((A \cup B) \cap C\)
1. Find \( A \cup B = \{1, 2, 3, 4, 5, 8\} \).
2. Find the intersection with \( C \): \( (A \cup B) \cap C = \{1, 2, 3, 5\} \).

#### (b) \(A \cup (B \cap C)\)
1. Find \( B \cap C = \{1, 2\} \).
2. Find the union with \( A \): \( A \cup (B \cap C) = \{1, 2, 3, 4, 5\} \).

#### (c) \( \overline{C} \cup D \)
1. Find the complement of \( C \) with respect to \( U \), \( \overline{C} = \{4, 6, 8, 9, 10\} \).
2. Find the union with \( D \): \( \overline{C} \cup D = \{2, 4, 6, 8, 9, 10\} \).

#### (d) \( C \cap D \)
1. Find the intersection of \( C \) and \( D \): \( C \cap D = \{2\} \).

#### (e) \((A \cup B) - C\)
1. Find \( A \cup B = \{1, 2, 3, 4, 5, 8\} \).
2. Subtract \( C \) from this union: \( (A \cup B) - C = \{4, 8\} \).

#### (f) \( A \cup (B - C) \)
1. Find \( B - C = \{4, 8\} \).
2. Find the union with \( A \): \( A \cup (B - C) = \{1, 2, 3, 4, 5, 8\} \).

#### (g) \((B - C) - D\)
1. From the previous part, \( B - C = \{4, 8\} \).
2. Subtract \( D \) from this result: \((B - C) - D = \emptyset\) (since both 4 and 8 are in \( D \)).

#### (h) \( B - (C - D) \)
1. Find \( C - D = \{1, 3, 5, 7\} \).
2. Subtract \( C - D \) from \( B \): \( B - (C - D) = \{2, 4, 8\} \).

#### (i) \((A \cup B) - (C \cap D)\)
1. From part (d), we know \( C \cap D = \{2\} \).
2. Find \( A \cup B = \{1, 2, 3, 4, 5, 8\} \).
3. Subtract \( C \cap D \) from this union: \((A \cup B) - (C \cap D) = \{1, 3, 4, 5, 8\}\).

### Summary of Answers
- (a) \((A \cup B) \cap C = \{1, 2, 3, 5\}\)
- (b) \(A \cup (B \cap C) = \{1, 2, 3, 4, 5\}\)
- (c) \( \overline{C} \cup D = \{2, 4, 6, 8, 9, 10\}\)
- (d) \( C \cap D = \{2\}\)
- (e) \((A \cup B) - C = \{4, 8\}\)
- (f) \( A \cup (B - C) = \{1, 2, 3, 4, 5, 8\}\)
- (g) \((B - C) - D = \emptyset\)
- (h) \( B - (C - D) = \{2, 4, 8\}\)
- (i) \((A \cup B) - (C \cap D) = \{1, 3, 4, 5, 8\}\)

Would you like further explanations on any of these solutions?

### Follow-up Questions
1. What would be the result if we add another element to set \( C \) that isn't in \( D \)?
2. How would the results change if we took the complement of \( D \) instead of \( C \)?
3. What if we apply different universal sets, such as \( U = \{1, 2, 3, \dots, 15\} \)?
4. How can these set operations be represented visually using Venn diagrams?
5. Can we apply De Morgan's laws to simplify any of these expressions?

### Tip
Remember, intersections narrow down to common elements, while unions combine all unique elements in set operations.

For U = {1, 2, 3, ..., 10}, let A = {1, 2, 3, 4, 5}, B = {1, 2, 4, 8}, C = {1, 2, 3, 5, 7}, and D = {2, 4, 6, 8}. Determine the elements of each of the following sets: (a) (A ∪ B) ∩ C, (b) A ∪ (B ∩ C), (c) C' ∪ D, (d) C ∩ D, (e) (A ∪ B) - C, (f) A ∪ (B - C), (g) (B - C) - D, (h) B - (C - D), (i) (A ∪ B) - (C ∩ D).

Solve a set theory problem involving sets A, B, C, and D within a universal set U = {1, 2, ..., 10}. This problem covers union, intersection, complement, and difference of sets, suitable for high school students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation