Let's break down and prove the two parts of the problem from the image.

Problem 24: Prove the following

Part (i):

$A \times (B \cup C) = (A \times B) \cup (A \times C)$

Proof: We will prove this by showing that each side of the equation represents the same set.

1. Left-hand side (LHS):
   $A \times (B \cup C)$ represents the Cartesian product of set $A$ with the union of sets $B$ and $C$. So: $A \times (B \cup C) = \{(a, x) \mid a \in A \text{ and } x \in (B \cup C)\}$ By definition of union, $x \in (B \cup C)$ means that $x \in B$ or $x \in C$. Therefore: $A \times (B \cup C) = \{(a, x) \mid a \in A \text{ and } (x \in B \text{ or } x \in C)\}$

2. Right-hand side (RHS):
   $(A \times B) \cup (A \times C)$ represents the union of two Cartesian products: $(A \times B) = \{(a, b) \mid a \in A \text{ and } b \in B\}$ $(A \times C) = \{(a, c) \mid a \in A \text{ and } c \in C\}$ Therefore, the union is: $(A \times B) \cup (A \times C) = \{(a, x) \mid a \in A \text{ and } (x \in B \text{ or } x \in C)\}$

Thus, we can see that both sides represent the same set. Hence,
$A \times (B \cup C) = (A \times B) \cup (A \times C)$

Part (ii):

$(A \cup B)^c = A^c \cap B^c$ where $A^c$ denotes the complement of $A$.

Proof: This is De Morgan's Law for set complements. We will prove it as follows:

1. Left-hand side (LHS):
   $(A \cup B)^c$ represents the complement of the union of $A$ and $B$. In set notation: $(A \cup B)^c = \{x \mid x \notin A \cup B\}$ That means $x$ is not in $A \cup B$, which implies that $x$ is neither in $A$ nor in $B$. Therefore: $(A \cup B)^c = \{x \mid x \notin A \text{ and } x \notin B\}$

2. Right-hand side (RHS):
   $A^c \cap B^c$ represents the intersection of the complements of $A$ and $B$: $A^c = \{x \mid x \notin A\}, \quad B^c = \{x \mid x \notin B\}$ Therefore, the intersection is: $A^c \cap B^c = \{x \mid x \notin A \text{ and } x \notin B\}$

Since both sides represent the same set, we have:
$(A \cup B)^c = A^c \cap B^c$

This concludes the proof for both parts.

Would you like further clarification or details on any step?

Here are 5 related questions for further practice:

1. Prove the distributive property $A \times (B \cap C) = (A \times B) \cap (A \times C)$.
2. Can you extend De Morgan's Law to three sets, i.e., prove $(A \cup B \cup C)^c = A^c \cap B^c \cap C^c$?
3. Show that $A \times (B - C) = (A \times B) - (A \times C)$.
4. Prove the general form of the union of Cartesian products, $(A_1 \cup A_2) \times B = (A_1 \times B) \cup (A_2 \times B)$.
5. If $A^c \cap B^c = \emptyset$, what can you say about $A \cup B$?

Tip: When working with set identities, using Venn diagrams can help visualize and understand relationships more clearly.