Given the sets S1 = {2, 4, 6}, S2 = {7, 2, 6}, S3 = {a, b, c} and U =

{1,2,3, ... ,10, a, b, c}, find:

a. S1 ∪ S2 b. S1 ∩ S2 c. S1 ∪ S3 d. S1 ∩ S3 e. S1

̃ (S1 Compliment) f.

S3

̃ (S3 Compliment)

Given the sets:

- \( S1 = \{2, 4, 6\} \)
- \( S2 = \{7, 2, 6\} \)
- \( S3 = \{a, b, c\} \)
- \( U = \{1, 2, 3, \dots, 10, a, b, c\} \)

Let's solve each part:

### a. \( S1 \cup S2 \) (Union of \(S1\) and \(S2\))
The union of two sets contains all the elements from both sets without duplicates.
\[
S1 \cup S2 = \{2, 4, 6\} \cup \{7, 2, 6\} = \{2, 4, 6, 7\}
\]

### b. \( S1 \cap S2 \) (Intersection of \(S1\) and \(S2\))
The intersection of two sets contains only the elements that are in both sets.
\[
S1 \cap S2 = \{2, 4, 6\} \cap \{7, 2, 6\} = \{2, 6\}
\]

### c. \( S1 \cup S3 \) (Union of \(S1\) and \(S3\))
\[
S1 \cup S3 = \{2, 4, 6\} \cup \{a, b, c\} = \{2, 4, 6, a, b, c\}
\]

### d. \( S1 \cap S3 \) (Intersection of \(S1\) and \(S3\))
Since \(S1\) and \(S3\) have no elements in common, the intersection is empty.
\[
S1 \cap S3 = \emptyset
\]

### e. \( \tilde{S1} \) (Complement of \(S1\) relative to \(U\))
The complement of \(S1\) contains all the elements in the universal set \(U\) that are not in \(S1\).
\[
\tilde{S1} = U - S1 = \{1, 2, 3, \dots, 10, a, b, c\} - \{2, 4, 6\} = \{1, 3, 5, 7, 8, 9, 10, a, b, c\}
\]

### f. \( \tilde{S3} \) (Complement of \(S3\) relative to \(U\))
The complement of \(S3\) contains all the elements in the universal set \(U\) that are not in \(S3\).
\[
\tilde{S3} = U - S3 = \{1, 2, 3, \dots, 10, a, b, c\} - \{a, b, c\} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}
\]

### Summary of Results:
- \( S1 \cup S2 = \{2, 4, 6, 7\} \)
- \( S1 \cap S2 = \{2, 6\} \)
- \( S1 \cup S3 = \{2, 4, 6, a, b, c\} \)
- \( S1 \cap S3 = \emptyset \)
- \( \tilde{S1} = \{1, 3, 5, 7, 8, 9, 10, a, b, c\} \)
- \( \tilde{S3} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)

Would you like more details on any part? 

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Here are 5 related questions:

1. What is the difference between union and intersection of sets?
2. How would you represent the complement of \(S2\) relative to \(U\)?
3. Can the complement of a set ever be empty? Under what conditions?
4. How do you represent the intersection of more than two sets?
5. What are some real-world applications of set theory?

**Tip**: When working with set complements, remember that the complement of a set is always relative to the universal set.

Given the sets S1 = {2, 4, 6}, S2 = {7, 2, 6}, S3 = {a, b, c} and U = {1,2,3, ... ,10, a, b, c}, find: a. S1 ∪ S2 b. S1 ∩ S2 c. S1 ∪ S3 d. S1 ∩ S3 e. S1̃ (S1 Compliment) f. S3̃ (S3 Compliment)

Explore set operations including union, intersection, and complement with sets S1, S2, S3, and universal set U. Learn how to find S1 ∪ S2, S1 ∩ S2, S1 ∪ S3, S1 ∩ S3, and the complements of S1 and S3 using set theory principles. Suitable for high school students in Grades 9-12.

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