You've uploaded an image with several math problems related to sets, Venn diagrams, and set operations. Let's go through the first problem together and then I'll provide solutions for the rest if needed.

Problem 1:

You're given the following sets:

• Universal set $U = \{1, 2, 3, ..., 11\}$
• $A = \{1, 3, 5, 7\}$
• $B = \{5, 9, 11\}$

Part a) Draw the Venn diagram based on the given sets.

• In the Venn diagram, you would have three regions: one for set $A$, one for set $B$, and the universal set $U$ outside of those. The common elements between $A$ and $B$ (i.e., the intersection $A \cap B$) would be placed in the overlap of $A$ and $B$.
  • Intersection: $A \cap B = \{5\}$
  • Elements in $A$ but not in $B$: $\{1, 3, 7\}$
  • Elements in $B$ but not in $A$: $\{9, 11\}$
  • Elements in $U$ but not in $A \cup B$: $\{2, 4, 6, 8, 10\}$

Part b) Calculate the cardinality of each set:

• $|A| = 4$ (since $A = \{1, 3, 5, 7\}$)
• $|B| = 3$ (since $B = \{5, 9, 11\}$)
• $|A \cup B| = 6$ (union of the sets: $A \cup B = \{1, 3, 5, 7, 9, 11\}$)
• $|A \cap B| = 1$ (intersection: $A \cap B = \{5\}$)

Would you like me to proceed with the rest of the problems? If yes, let me know if you'd prefer detailed explanations or just final answers.

Here are 5 questions related to the above problem:

1. How would the Venn diagram change if $B$ were a subset of $A$?
2. What is the complement of $A$ in the universal set $U$?
3. How would the cardinality of $A \cup B$ change if the sets were disjoint?
4. How can you compute $A \setminus B$ (elements in $A$ but not in $B$)?
5. What is the relationship between $|A|$, $|B|$, and $|A \cap B|$ in terms of union?

Tip: When drawing Venn diagrams, always start by identifying the intersections of sets to ensure clarity in the diagram layout.