The problem consists of two parts. The first part asks us to simplify the given set expressions using the laws of sets. The second part asks us to describe the given sets using set builder notation.

1. Problem Description

The problem consists of two parts. The first part asks us to simplify the given set expressions using the laws of sets. The second part asks us to describe the given sets using set builder notation.

2. Solution Steps

Part 3: Simplifying Set Expressions

(a) $[A' \cup (B \cap C)]'$

Using De Morgan's Law: $(X \cup Y)' = X' \cap Y'$, we have

$[A' \cup (B \cap C)]' = (A')' \cap (B \cap C)'$

Since $(A')' = A$, we have

$A \cap (B \cap C)'$

Using De Morgan's Law: $(X \cap Y)' = X' \cup Y'$, we have

$A \cap (B' \cup C')$

Using the distributive law: $X \cap (Y \cup Z) = (X \cap Y) \cup (X \cap Z)$, we have

$(A \cap B') \cup (A \cap C')$

(b) $(A - B) \cup (A \cap B)$

$A - B$ is defined as $A \cap B'$. Thus, the expression becomes:

$(A \cap B') \cup (A \cap B)$

Using the distributive law: $X \cap (Y \cup Z) = (X \cap Y) \cup (X \cap Z)$ in reverse, we have

$A \cap (B' \cup B)$

Since $B' \cup B = U$ (the universal set), we have

$A \cap U = A$

(c) $(A - B)'$

$A - B$ is defined as $A \cap B'$. Thus, the expression becomes:

$(A \cap B')'$

Using De Morgan's Law: $(X \cap Y)' = X' \cup Y'$, we have

$A' \cup (B')' = A' \cup B$

(d) $(A \cup B) \cap (A \cup B')$

Using the distributive law: $X \cup (Y \cap Z) = (X \cup Y) \cap (X \cup Z)$ in reverse, we have

$A \cup (B \cap B')$

Since $B \cap B' = \emptyset$ (the empty set), we have

$A \cup \emptyset = A$

Part 4: Describing Sets in Set Builder Notation

(a) $A = \{1, 8, 27, 64, 125\}$

The elements are $1^3, 2^3, 3^3, 4^3, 5^3$. Thus,

$A = \{x \mid x = n^3, n \in \{1, 2, 3, 4, 5\}\}$

(b) $B = \{0, 2, 4, 6, 8, 12\}$

The elements are even numbers. We can write $B = \{x \mid x = 2n, n \in \{0, 1, 2, 3, 4, 6\}\}$.

Alternatively, we can write $B = \{x \mid x \text{ is an even integer and } 0 \le x \le 12, x \ne 10 \}$

(c) $C = \{1, 3, 9, 27, 81\}$

The elements are powers of 3: $3^0, 3^1, 3^2, 3^3, 3^4$. Thus,

$C = \{x \mid x = 3^n, n \in \{0, 1, 2, 3, 4\}\}$

(d) $D = \{(1, 2), (2, 4), (3, 6), (4, 8)\}$

Each element is an ordered pair $(x, y)$ where $y = 2x$.

$D = \{(x, y) \mid y = 2x, x \in \{1, 2, 3, 4\}\}$

3. Final Answer

3. (a) $(A \cap B') \cup (A \cap C')$

(b) $A$

(c) $A' \cup B$

(d) $A$

4. (a) $A = \{x \mid x = n^3, n \in \{1, 2, 3, 4, 5\}\}$

(b) $B = \{x \mid x = 2n, n \in \{0, 1, 2, 3, 4, 6\}\}$

(c) $C = \{x \mid x = 3^n, n \in \{0, 1, 2, 3, 4\}\}$

(d) $D = \{(x, y) \mid y = 2x, x \in \{1, 2, 3, 4\}\}$