Let U = Universal set ={
 a,b,c,d,e,1,2,3
 }
 , A={2,1,3,a,b}
 ,   and B={c,1,d}
.
List the elements of the following sets. If there is more than one element write them separated by commas.
A¯¯¯¯={
 
c,d,e
 }

A∪B¯¯¯¯¯¯¯¯¯¯¯¯¯={
 
1, 2, 3, a, b, e 
 }

A¯¯¯¯∩B¯¯¯¯={
 
e
 }

Let's break down each set operation step by step:

### Universal Set (U):
\( U = \{a, b, c, d, e, 1, 2, 3\} \)

### Given sets:
- \( A = \{2, 1, 3, a, b\} \)
- \( B = \{c, 1, d\} \)

### 1. \( \overline{A} \) (Complement of A with respect to U):

The complement of \( A \) is the set of all elements in the universal set \( U \) that are not in \( A \).

\[
\overline{A} = U - A = \{c, d, e\}
\]

### 2. \( A \cup \overline{B} \) (Union of A and the complement of B):

First, find the complement of \( B \) with respect to \( U \):
\[
\overline{B} = U - B = \{a, b, e, 2, 3\}
\]

Now, take the union of \( A \) and \( \overline{B} \):
\[
A \cup \overline{B} = \{2, 1, 3, a, b\} \cup \{a, b, e, 2, 3\} = \{1, 2, 3, a, b, e\}
\]

### 3. \( \overline{A} \cap \overline{B} \) (Intersection of the complements of A and B):

We already found:
- \( \overline{A} = \{c, d, e\} \)
- \( \overline{B} = \{a, b, e, 2, 3\} \)

Now, take the intersection:
\[
\overline{A} \cap \overline{B} = \{c, d, e\} \cap \{a, b, e, 2, 3\} = \{e\}
\]

### Final Answers:

1. \( \overline{A} = \{c, d, e\} \)
2. \( A \cup \overline{B} = \{1, 2, 3, a, b, e\} \)
3. \( \overline{A} \cap \overline{B} = \{e\} \)

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What is the complement of \( B \) if \( U \) is a different set?
2. How do you find the union of three sets?
3. What is the symmetric difference of sets \( A \) and \( B \)?
4. How would the result change if \( A \cap B \) was given?
5. How do you represent set operations using Venn diagrams?

**Tip**: When calculating the complement of a set, always check which elements belong to the universal set but not to the specific set in question.

Let U = Universal set = {a,b,c,d,e,1,2,3}, A = {2,1,3,a,b}, and B = {c,1,d}. List the elements of the following sets: A¯ = {c,d,e}, A ∪ B¯ = {1, 2, 3, a, b, e}, A¯ ∩ B¯ = {e}.

Explore how to solve set theory problems involving the universal set U = {a,b,c,d,e,1,2,3}, A = {2,1,3,a,b}, and B = {c,1,d}. Learn about the complement, union, and intersection of sets with detailed solutions and mathematical formulas. Ideal for students in Grades 9-12.

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