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¯¯¯¯¯¯¯¯¯¯¯¯¯={
 
 }

A¯¯¯¯∩B¯¯¯¯={
 
e
 }

Let's analyze each set step by step:

1. **Complement of A (denoted as \( \overline{A} \)):**
   The complement of A includes all the elements of the Universal set \( U \) that are **not** in set \( A \).

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

   So, \( \overline{A} = U - A = \{c, d, e\} \)

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

2. **Union of \( A \) and complement of \( B \) (denoted as \( A \cup \overline{B} \)):**
   The complement of \( B \) includes all the elements of \( U \) that are **not** in set \( B \).

   Set \( B = \{c, 1, d\} \)

   So,  
   \[
   \overline{B} = U - B = \{a, b, e, 2, 3\}
   \]

   Now, the union of \( A \) and \( \overline{B} \) is the set of all elements that are in either \( A \) or \( \overline{B} \), or in both:

   \[
   A \cup \overline{B} = \{2, 1, 3, a, b\} \cup \{a, b, e, 2, 3\} = \{1, 2, 3, a, b, e\}
   \]

   Therefore,  
   \[
   A \cup \overline{B} = \{1, 2, 3, a, b, e\}
   \]

3. **Intersection of \( \overline{A} \) and \( \overline{B} \) (denoted as \( \overline{A} \cap \overline{B} \)):**
   The intersection includes elements that are in both \( \overline{A} \) and \( \overline{B} \).

   From step 1,  
   \( \overline{A} = \{c, d, e\} \)  
   From step 2,  
   \( \overline{B} = \{a, b, e, 2, 3\} \)

   The common element in both sets is \( e \).

   Therefore,  
   \[
   \overline{A} \cap \overline{B} = \{e\}
   \]

### Final Results:
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 further details or explanations on any of these? Here are 5 questions related to this topic to help you expand your understanding:

1. What is the complement of a set, and how is it calculated?
2. How do we find the union of two sets?
3. What is the intersection of two sets and how is it determined?
4. Can a complement and union operation ever result in the same set?
5. How does the universal set affect the complement of subsets within it?

**Tip:** Always start by finding the complement of sets when dealing with complement operations. This simplifies the process of solving complex set problems.

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¯¯¯¯¯¯¯¯¯¯¯¯¯={}, A¯¯¯¯∩B¯¯¯¯={e}.

Learn how to solve a set theory problem involving the universal set U={a,b,c,d,e,1,2,3}, A={2,1,3,a,b}, and B={c,1,d}. This example covers complements, union, and intersection of sets with step-by-step explanations. Suitable for Grades 9-10.

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