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.