집합과 명제 전체 문제

Let $U = \left\{ {1,2,3,4,5,6,7,8,9,10} \right\}$, and let

$$\begin{aligned}A & = \left\{ {3,4,5,6,7} \right\}\!, & B & = \left\{ {1,5,7,9} \right\}\!, \\ C & = \left\{ {3,6,9} \right\}\!, & D & = \left\{ {2,4,6,8} \right\}\!\end{aligned}$$

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

$B - D$

Let $U = \left\{ {1,2,3,4,5,6,7,8,9,10} \right\}$, and let

$$\begin{aligned}A & = \left\{ {3,4,5,6,7} \right\}\!, & B & = \left\{ {1,5,7,9} \right\}\!, \\ C & = \left\{ {3,6,9} \right\}\!, & D & = \left\{ {2,4,6,8} \right\}\!\end{aligned}$$

. 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 $U = \left\{ {1,2,3,4,5,6,7,8,9,10} \right\}$, and let

$$\begin{aligned}A & = \left\{ {3,4,5,6,7} \right\}\!, & B & = \left\{ {1,5,7,9} \right\}\!, \\ C & = \left\{ {3,6,9} \right\}\!, & D & = \left\{ {2,4,6,8} \right\}\!\end{aligned}$$

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

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

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.

$A \cap B$

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.

$A \cup B$

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.

$\left( {A \cup B} \right)^c$

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.

$A^c \cap B^c$

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.

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

Let $P$, $Q$, $R$, and $S$ be subsets of a universal set $U$. Assume that $\left( P - Q \right) \subseteq \left( R \cap S \right)$.

Complete the following sentence:

For each

x \in U

, if

x \in \left( P - Q \right)

, then

\ldots

.

Let $U$ be the universal set. Consider the following statement:

For all

A

and

B

that are subsets of

U

, if

A \subseteq B

, then

B^c \subseteq A^c

.

Identify three conditional statements in the given statement.

Let $U$ be the universal set. Consider the following statement:

For all

A

and

B

that are subsets of

U

, if

A \subseteq B

, then

B^c \subseteq A^c

.

Write the contrapositive of this statement.

Let $U$ be the universal set. Consider the following statement:

For all

A

and

B

that are subsets of

U

, if

A \subseteq B

, then

B^c \subseteq A^c

.

Write the negation of this statement.

Let $A$, $B$, and $C$ be subsets of some universal set $U$. Draw a Venn diagram for each of the following situations.

$A \subseteq C$