집합과 명제 전체 문제

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.

$A \subseteq C$

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

$A \cap B = \emptyset$

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

$A \not \subseteq B,B \not \subseteq A,C \subseteq A,\text{ and } C \not \subseteq B$

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

$A \subseteq B,C \subseteq B,\text{ and } A \cap C = \emptyset$

Let $A$, $B$, and $C$ be subsets of some universal set $U$. For each of the following, draw a general Venn diagram for the three sets and then shade the indicated region.

$A \cap B$

Let $A$, $B$, and $C$ be subsets of some universal set $U$. For each of the following, draw a general Venn diagram for the three sets and then shade the indicated region.

$A \cap C$

Let $A$, $B$, and $C$ be subsets of some universal set $U$. For each of the following, draw a general Venn diagram for the three sets and then shade the indicated region.

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

Let $A$, $B$, and $C$ be subsets of some universal set $U$. For each of the following, draw a general Venn diagram for the three sets and then shade the indicated region.

$B \cup C$

Let $A$, $B$, and $C$ be subsets of some universal set $U$. For each of the following, draw a general Venn diagram for the three sets and then shade the indicated region.

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

Let $A$, $B$, and $C$ be subsets of some universal set $U$. For each of the following, draw a general Venn diagram for the three sets and then shade the indicated region.

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

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

Draw a Venn diagram with $A \subseteq B$ and $B \subseteq C$. Does it appear that $A \subseteq C$?

Let $S$, $T$, $X$, and $Y$ be subsets of some universal set. Assume that

• (i) $S \cup T \subseteq X \cup Y$; - (ii) $S \cap T = \emptyset$; and - (iii) $X \subseteq S$.

Using assumption (i), what conclusion(s) can be made if it is known that $a \in T$?

Let $S$, $T$, $X$, and $Y$ be subsets of some universal set. Assume that

• (i) $S \cup T \subseteq X \cup Y$; - (ii) $S \cap T = \emptyset$; and - (iii) $X \subseteq S$.

Using assumption (ii), what conclusion(s) can be made if it is known that $a \in T$?

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