Answer:
Let's start by defining the sets:
- \( U \) (Universal set) = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}
- \( A \) (Odd numbers) = \{1, 3, 5, 7, 9, 11\}
- \( B \) (Even numbers) = \{2, 4, 6, 8, 10, 12\}
- \( C \) (Prime numbers) = \{2, 3, 5, 7, 11\}
Now, let's create the Venn diagram and answer the questions:
1. **\( A \cup B \) (A union B)**: This set includes all elements that are in \( A \) or \( B \) or both.
- \( A \cup B \) = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}
- Since every number in the universal set is either odd or even, \( A \cup B = U \).
2. **\( A \cup C \) (A union C)**: This set includes all elements that are in \( A \) or \( C \) or both.
- \( A \cup C \) = \{1, 3, 5, 7, 9, 11, 2\}
3. **\( B \cup C \) (B union C)**: This set includes all elements that are in \( B \) or \( C \) or both.
- \( B \cup C \) = \{2, 4, 6, 8, 10, 12, 3, 5, 7, 11\}
4. **\( A \cap B \) (A intersection B)**: This set includes all elements that are in both \( A \) and \( B \).
- \( A \cap B \) = \{\} (There are no numbers that are both odd and even)
5. **\( A \cap C \) (A intersection C)**: This set includes all elements that are in both \( A \) and \( C \).
- \( A \cap C \) = \{3, 5, 7, 11\}
6. **\( B \cap C \) (B intersection C)**: This set includes all elements that are in both \( B \) and \( C \).
- \( B \cap C \) = \{2\}
Here's the Venn diagram:
```
_______A_______
/ \
/ {1,3,5,7,9,11} \
/ \
/ \
/ _______C_______ \
/ / {2,3,5,7,11} \ \
/ / \ \
/ / \ \
/ / B \ \
/ / {2,4,6,8,10,12} \ \
/ /__________________________\ \
/______________________________________\
```
From the Venn diagram, you can clearly see the sets and their intersections.
