Let's define the sets based on the problem:

• $S = \{ \text{FF}, \text{FM}, \text{MF}, \text{MM} \}$ is the set of all possible gender combinations of two children.
• $A$ is the subset containing no males.
• $B$ is the subset containing two males.
• $C$ is the subset containing at least one male.

Now, let's determine each of the subsets:

1. Set $A$ (No males):

   • $A = \{ \text{FF} \}$
   • This set contains only the pair where both children are female.

2. Set $B$ (Two males):

   • $B = \{ \text{MM} \}$
   • This set contains only the pair where both children are male.

3. Set $C$ (At least one male):

   • $C = \{ \text{FM}, \text{MF}, \text{MM} \}$
   • This set contains all pairs where at least one child is male.

Next, let's determine the intersections and unions of these sets:

4. Intersection $A \cap B$:

   • $A \cap B = \{ \text{FF} \} \cap \{ \text{MM} \} = \emptyset$
   • No elements are common between $A$ and $B$, so the intersection is the empty set.

5. Union $A \cup B$:

   • $A \cup B = \{ \text{FF} \} \cup \{ \text{MM} \} = \{ \text{FF}, \text{MM} \}$
   • The union contains the elements from both sets $A$ and $B$.

6. Intersection $A \cap C$:

   • $A \cap C = \{ \text{FF} \} \cap \{ \text{FM}, \text{MF}, \text{MM} \} = \emptyset$
   • $A$ (no males) has no common elements with $C$ (at least one male), so the intersection is the empty set.

7. Union $A \cup C$:

   • $A \cup C = \{ \text{FF} \} \cup \{ \text{FM}, \text{MF}, \text{MM} \} = \{ \text{FF}, \text{FM}, \text{MF}, \text{MM} \}$
   • The union covers all possible cases, so it equals the universal set $S$.

8. Intersection $B \cap C$:

   • $B \cap C = \{ \text{MM} \} \cap \{ \text{FM}, \text{MF}, \text{MM} \} = \{ \text{MM} \}$
   • The only common element between $B$ (two males) and $C$ (at least one male) is $\text{MM}$.

9. Union $B \cup C$:

   • $B \cup C = \{ \text{MM} \} \cup \{ \text{FM}, \text{MF}, \text{MM} \} = \{ \text{FM}, \text{MF}, \text{MM} \}$
   • The union contains all pairs where at least one male is present, which is the same as $C$.

10. Intersection $C \cap B$:

    • $C \cap B = \{ \text{FM}, \text{MF}, \text{MM} \} \cap \{ \text{MM} \} = \{ \text{MM} \}$
    • As noted earlier, this is just the $MM$ pair.

Summary of the Sets:

• $A = \{ \text{FF} \}$
• $B = \{ \text{MM} \}$
• $C = \{ \text{FM}, \text{MF}, \text{MM} \}$
• $A \cap B = \emptyset$
• $A \cup B = \{ \text{FF}, \text{MM} \}$
• $A \cap C = \emptyset$
• $A \cup C = \{ \text{FF}, \text{FM}, \text{MF}, \text{MM} \}$
• $B \cap C = \{ \text{MM} \}$
• $B \cup C = \{ \text{FM}, \text{MF}, \text{MM} \}$

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Relative Questions

1. What is the principle behind set operations like union and intersection in probability?
2. How do complement sets affect the calculations in problems involving multiple groups?
3. Why is it important to consider all possible outcomes in a set-theory problem?
4. How can set theory be applied to more complex scenarios with more than two groups?
5. What is the significance of the empty set in problems involving intersections?

Tip:

When working with sets, drawing a Venn diagram can greatly help in visualizing the relationships and intersections between different sets.