집합의 연산

Let $U = \mathbb{N}$, and let

$$\begin{aligned}A & = \left\{ x \in \mathbb{N} \mid x \geq 7 \right\}\!, & B & = \left\{ x \in \mathbb{N} \mid x \text{ is odd } \right\}\!, \\ C & = \left\{ x \in \mathbb{N} \mid x \text{ is a multiple of 3 } \right\}\!, & D & = \left\{ x \in \mathbb{N} \mid x \text{ is even } \right\}\!\end{aligned}$$

. Use the roster method to list all of the elements of each of the following sets.

$A^c \cap B^c$

Let $U = \mathbb{N}$, and let

$$\begin{aligned}A & = \left\{ x \in \mathbb{N} \mid x \geq 7 \right\}\!, & B & = \left\{ x \in \mathbb{N} \mid x \text{ is odd } \right\}\!, \\ C & = \left\{ x \in \mathbb{N} \mid x \text{ is a multiple of 3 } \right\}\!, & D & = \left\{ x \in \mathbb{N} \mid x \text{ is even } \right\}\!\end{aligned}$$

. Use the roster method to list all of the elements of each of the following sets.

$\left( {A \cup B} \right) \cap C$

Let $U = \mathbb{N}$, and let

$$\begin{aligned}A & = \left\{ x \in \mathbb{N} \mid x \geq 7 \right\}\!, & B & = \left\{ x \in \mathbb{N} \mid x \text{ is odd } \right\}\!, \\ C & = \left\{ x \in \mathbb{N} \mid x \text{ is a multiple of 3 } \right\}\!, & D & = \left\{ x \in \mathbb{N} \mid x \text{ is even } \right\}\!\end{aligned}$$

. Use the roster method to list all of the elements of each of the following sets.

$\left( {A \cap C} \right) \cup \left( {B \cap C} \right)$

Let $U = \mathbb{N}$, and let

$$\begin{aligned}A & = \left\{ x \in \mathbb{N} \mid x \geq 7 \right\}\!, & B & = \left\{ x \in \mathbb{N} \mid x \text{ is odd } \right\}\!, \\ C & = \left\{ x \in \mathbb{N} \mid x \text{ is a multiple of 3 } \right\}\!, & D & = \left\{ x \in \mathbb{N} \mid x \text{ is even } \right\}\!\end{aligned}$$

. Use the roster method to list all of the elements of each of the following sets.

$B \cap D$

Let $U = \mathbb{N}$, and let

$$\begin{aligned}A & = \left\{ x \in \mathbb{N} \mid x \geq 7 \right\}\!, & B & = \left\{ x \in \mathbb{N} \mid x \text{ is odd } \right\}\!, \\ C & = \left\{ x \in \mathbb{N} \mid x \text{ is a multiple of 3 } \right\}\!, & D & = \left\{ x \in \mathbb{N} \mid x \text{ is even } \right\}\!\end{aligned}$$

. Use the roster method to list all of the elements of each of the following sets.

$\left( {B \cap D} \right)^c$

Let $U = \mathbb{N}$, and let

$$\begin{aligned}A & = \left\{ x \in \mathbb{N} \mid x \geq 7 \right\}\!, & B & = \left\{ x \in \mathbb{N} \mid x \text{ is odd } \right\}\!, \\ C & = \left\{ x \in \mathbb{N} \mid x \text{ is a multiple of 3 } \right\}\!, & D & = \left\{ x \in \mathbb{N} \mid x \text{ is even } \right\}\!\end{aligned}$$

. Use the roster method to list all of the elements of each of the following sets.

$A - D$

Let $U = \mathbb{N}$, and let

$$\begin{aligned}A & = \left\{ x \in \mathbb{N} \mid x \geq 7 \right\}\!, & B & = \left\{ x \in \mathbb{N} \mid x \text{ is odd } \right\}\!, \\ C & = \left\{ x \in \mathbb{N} \mid x \text{ is a multiple of 3 } \right\}\!, & D & = \left\{ x \in \mathbb{N} \mid x \text{ is even } \right\}\!\end{aligned}$$

. Use the roster method to list all of the elements of each of the following sets.

$B - D$

Let $U = \mathbb{N}$, and let

$$\begin{aligned}A & = \left\{ x \in \mathbb{N} \mid x \geq 7 \right\}\!, & B & = \left\{ x \in \mathbb{N} \mid x \text{ is odd } \right\}\!, \\ C & = \left\{ x \in \mathbb{N} \mid x \text{ is a multiple of 3 } \right\}\!, & D & = \left\{ x \in \mathbb{N} \mid x \text{ is even } \right\}\!\end{aligned}$$

. Use the roster method to list all of the elements of each of the following sets.

$\left( A - D \right) \cup \left( B - D \right)$

Let $U = \mathbb{N}$, and let

$$\begin{aligned}A & = \left\{ x \in \mathbb{N} \mid x \geq 7 \right\}\!, & B & = \left\{ x \in \mathbb{N} \mid x \text{ is odd } \right\}\!, \\ C & = \left\{ x \in \mathbb{N} \mid x \text{ is a multiple of 3 } \right\}\!, & D & = \left\{ x \in \mathbb{N} \mid x \text{ is even } \right\}\!\end{aligned}$$

. Use the roster method to list all of the elements of each of the following sets.

$\left( A \cup B \right) - D$

Draw two general Venn diagrams for the sets $A$, $B$, and $C$. On one, shade the region that represents $\left( {A \cup B} \right) - C$, and on the other, shade the region that represents $\left( {A - C} \right) \cup \left( {B - C} \right)$. Explain carefully how you determined these regions and why they indicate that $\left( {A \cup B} \right) - C = \left( {A - C} \right) \cup \left( {B - C} \right)$.

Let $A$, $B$, and $C$ be subsets of some universal set $U$.

Draw two general Venn diagrams for the sets $A$, $B$, and $C$. On one, shade the region that represents $A - \left( {B \cup C} \right)$, and on the other, shade the region that represents $\left( {A - B} \right) \cap \left( {A - C} \right)$. Based on the Venn diagrams, make a conjecture about the relationship between the sets $A - \left( {B \cup C} \right)$ and $\left( {A - B} \right) \cap \left( {A - C} \right)$.

Let $A$, $B$, and $C$ be subsets of some universal set $U$.

Draw two general Venn diagrams for the sets $A$, $B$, and $C$. On one, shade the region that represents $A - \left( {B \cap C} \right)$, and on the other, shade the region that represents $\left( {A - B} \right) \cup \left( {A - C} \right)$. Based on the Venn diagrams, make a conjecture about the relationship between the sets $A - \left( {B \cap C} \right)$ and $\left( {A - B} \right) \cup \left( {A - C} \right)$.

Let $A$, $B$, and $C$ be subsets of some universal set $U$.

Draw two general Venn diagrams for the sets $A$, $B$, and $C$. On one, shade the region that represents $A - \left( {B - C} \right)$, and on the other, shade the region that represents $\left( {A - B} \right) - C$. Based on the Venn diagrams, make a conjecture about the relationship between the sets $A - \left( {B - C} \right)$ and $\left( {A - B} \right) - C$. (Are the two sets equal? If not, is one of the sets a subset of the other set?)

Let $A$, $B$, and $C$ be subsets of some universal set $U$.

Draw two general Venn diagrams for the sets $A$, $B$, and $C$. On one, shade the region that represents $A - \left( {B - C} \right)$, and on the other, shade the region that represents $\left( {A - B} \right) \cup \left( A - C^c \right)$. Based on the Venn diagrams, make a conjecture about the relationship between the sets $A - \left( {B - C} \right)$ and $\left( {A - B} \right) \cup \left( A - C^c \right)$. (Are the two sets equal? If not, is one of the sets a subset of the other set?)