The image shows a problem involving set theory, where you are asked to simplify three expressions involving sets $A$, $B$, and $C$ (subsets of some universal set $\Omega$) and to illustrate them using Venn diagrams. The expressions are:

1. $(A \cup B) \cap (A \cup B^c)$
2. $(A \cup B) \cap (A \cup C)$
3. $(A^c \cap B^c) \cup (B^c \cap C^c)$

Let's simplify each expression step by step.

1. $(A \cup B) \cap (A \cup B^c)$

• Simplification:
  • $A \cup B$ is the set of elements in $A$, $B$, or both.
  • $A \cup B^c$ is the set of elements in $A$ or not in $B$.
  • The intersection $(A \cup B) \cap (A \cup B^c)$ includes elements in both sets:
    • This intersection is equivalent to $A$ because any element in $A$ will be in both $A \cup B$ and $A \cup B^c$.
• Result: $A$

2. $(A \cup B) \cap (A \cup C)$

• Simplification:
  • $A \cup B$ is the set of elements in $A$, $B$, or both.
  • $A \cup C$ is the set of elements in $A$, $C$, or both.
  • The intersection $(A \cup B) \cap (A \cup C)$ includes elements that are in either $A$, or in both $B$ and $C$.
• Result: $A \cup (B \cap C)$

3. $(A^c \cap B^c) \cup (B^c \cap C^c)$

• Simplification:
  • $A^c \cap B^c$ is the set of elements not in $A$ and not in $B$.
  • $B^c \cap C^c$ is the set of elements not in $B$ and not in $C$.
  • The union $(A^c \cap B^c) \cup (B^c \cap C^c)$ covers elements not in $B$ and either not in $A$ or not in $C$.
• Result: $B^c \cap (A^c \cup C^c)$, which is $B^c \cap (A \cap C)^c$.

Venn Diagrams:

To illustrate these, you can create a Venn diagram with three overlapping circles representing sets $A$, $B$, and $C$, and then shade the regions corresponding to the simplified expressions.

Would you like further details or any specific questions answered? Here are some additional questions related to this topic:

1. How can you generalize the simplification of set expressions for other combinations?
2. What would be the result if the sets were disjoint?
3. Can you prove the results using De Morgan's laws?
4. How would the results change if additional set operations like symmetric difference were introduced?
5. How can you apply these principles in probability theory?

Tip: Remember that when simplifying set expressions, identifying common elements and using basic set identities (like De Morgan's laws or distributive properties) can significantly streamline the process.