The image contains a problem set involving sets and Venn diagrams. Here's the breakdown of the problem:

Given sets:

• $U = \{1, 2, 3, 4, \dots, 25\}$
• $A = \{2, 4, 6, 8, 10, 12, 14, 16\}$
• $B = \{1, 3, 5, 7, 9, 11, 13, 15, 17\}$
• $C = \{1, 3, 5, 7, 11, 17\}$

You are asked to:

1. Draw a Venn diagram of these sets.
2. Determine the following operations:
   • (a) $U \cap A$
   • (b) $B \cap C$
   • (c) $A \cup C$
   • (d) $B - C$
   • (e) $U - A$
   • (f) $\overline{A} \cap \overline{B}$ (complement of A and B)
   • (g) $B \oplus C$ (symmetric difference of B and C)
   • (h) The cardinalities $|A|, |B|, |C|$
   • (i) The probabilities $P(A), P(C)$ (assuming probability is based on set size divided by universal set size).

Let's solve each part:

1. (a) $U \cap A$:

   • The universal set $U$ contains all elements from 1 to 25, and $A$ is a subset of $U$.
   • So, $U \cap A = A = \{2, 4, 6, 8, 10, 12, 14, 16\}$.

2. (b) $B \cap C$:

   • Intersection of $B$ and $C$ are the elements common to both sets.
   • $B \cap C = \{1, 3, 5, 7, 11, 17\}$ (since $C$ is a subset of $B$).

3. (c) $A \cup C$:

   • The union includes all elements from $A$ and $C$ without repetition.
   • $A \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 16, 17\}$.

4. (d) $B - C$:

   • This is the set of elements in $B$ that are not in $C$.
   • $B - C = \{9, 13, 15\}$.

5. (e) $U - A$:

   • Elements in $U$ but not in $A$.
   • $U - A = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25\}$.

6. (f) $\overline{A} \cap \overline{B}$:

   • Complement of $A$ in $U$ is $\overline{A} = U - A$, and complement of $B$ in $U$ is $\overline{B} = U - B$.
   • $\overline{A} = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25\}$
   • $\overline{B} = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 19, 20, 21, 22, 23, 24, 25\}$
   • $\overline{A} \cap \overline{B} = \{18, 19, 20, 21, 22, 23, 24, 25\}$.

7. (g) $B \oplus C$:

   • Symmetric difference includes elements in either $B$ or $C$, but not in both.
   • $B \oplus C = B \cup C - (B \cap C) = \{9, 13, 15\}$.

8. (h) $|A|, |B|, |C|$:

   • The cardinality (number of elements) in each set:
     • $|A| = 8$ (since $A = \{2, 4, 6, 8, 10, 12, 14, 16\}$),
     • $|B| = 9$ (since $B = \{1, 3, 5, 7, 9, 11, 13, 15, 17\}$),
     • $|C| = 6$ (since $C = \{1, 3, 5, 7, 11, 17\}$).

9. (i) $P(A), P(C)$:

   • Assuming the probability is based on the ratio of elements to the total size of $U$ (which contains 25 elements),
     • $P(A) = \frac{|A|}{|U|} = \frac{8}{25}$,
     • $P(C) = \frac{|C|}{|U|} = \frac{6}{25}$.

Would you like more details or a specific explanation on any part?

Related Questions:

1. What is the union $B \cup C$?
2. How do you interpret $A \oplus B$ (symmetric difference between $A$ and $B$)?
3. What would be the result of $U \cap B$?
4. Can we find the complement of $C$ in $U$?
5. What happens when you take the difference $C - B$?

Tip:

When working with set operations, always consider if you're asked for union, intersection, or difference—each gives different types of information about the relationship between sets.