1. **Problem Statement:** Given sets: $$A = \{x : x \in \mathbb{N} \text{ and } x \text{ is even}\},$$ $$B = \{x : x \in \mathbb{N} \text{ and } x \text{ is prime}\},$$ $$C = \{x : x \in \mathbb{N} \text{ and } x \text{ is a multiple of } 5\}.$$ Describe the following sets: (a) $A \cap B$ (intersection of $A$ and $B$) (b) $B \cap C$ (intersection of $B$ and $C$) (c) $A \cup B$ (union of $A$ and $B$) (d) $A \cap (B \cup C)$ (intersection of $A$ and union of $B$ and $C$) 2. **Solution for (a):** - The intersection $A \cap B$ contains elements that are both even and prime. - The only even prime number is 2. - Therefore, $$A \cap B = \{2\}.$$ 3. **Solution for (b):** - The intersection $B \cap C$ contains elements that are both prime and multiples of 5. - The only prime multiple of 5 is 5 itself. - Therefore, $$B \cap C = \{5\}.$$ 4. **Solution for (c):** - The union $A \cup B$ contains all elements that are either even or prime. - This set includes all even natural numbers and all prime numbers. - So, $$A \cup B = \{x : x \in \mathbb{N} \text{ and } (x \text{ is even or } x \text{ is prime})\}.$$ 5. **Solution for (d):** - The union $B \cup C$ contains all elements that are prime or multiples of 5. - The intersection $A \cap (B \cup C)$ contains elements that are even and also either prime or multiples of 5. - Since the only even prime is 2 and multiples of 5 that are even are multiples of 10, this set is: $$A \cap (B \cup C) = \{2\} \cup \{x : x \text{ is multiple of } 10\}.$$ 6. **Proof of Set Equality (2i):** Prove: $$A \cup (B \cap C) = (A \cup B) \cap (A \cup C).$$ - To prove two sets are equal, show each is a subset of the other. - **Inclusion One:** Show $$A \cup (B \cap C) \subseteq (A \cup B) \cap (A \cup C).$$ - If $x \in A$, then $x \in A \cup B$ and $x \in A \cup C$, so $x \in (A \cup B) \cap (A \cup C)$. - If $x \in B \cap C$, then $x \in B$ and $x \in C$, so $x \in A \cup B$ and $x \in A \cup C$. - Hence, $x \in (A \cup B) \cap (A \cup C)$. - **Inclusion Two:** Show $$(A \cup B) \cap (A \cup C) \subseteq A \cup (B \cap C).$$ - If $x \in (A \cup B) \cap (A \cup C)$, then $x \in A \cup B$ and $x \in A \cup C$. - If $x \in A$, then $x \in A \cup (B \cap C)$. - If $x \notin A$, then $x \in B$ and $x \in C$, so $x \in B \cap C$. - Thus, $x \in A \cup (B \cap C)$. - Since both inclusions hold, the sets are equal. 7. **Proof of Set Equality (2ii):** Prove: $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C).$$ - This is the distributive law of sets. - An element in $A \cap (B \cup C)$ is in $A$ and also in $B$ or $C$. - So it must be in either $A \cap B$ or $A \cap C$. - Conversely, any element in $(A \cap B) \cup (A \cap C)$ is in $A$ and in $B$ or $C$. - Hence, the sets are equal. **Final answers:** - (a) $A \cap B = \{2\}$ - (b) $B \cap C = \{5\}$ - (c) $A \cup B = \{x : x \in \mathbb{N} \text{ and } (x \text{ is even or prime})\}$ - (d) $A \cap (B \cup C) = \{2\} \cup \{x : x \text{ is multiple of } 10\}$ - Proofs for equalities (2i) and (2ii) shown above.