집합의 뜻과 포함 관계

Assume that the universal set is the set of integers. Let

$$\begin{aligned}A & = \left\{ { - 3, - 2,2,3} \right\}\!, & B & = \left\{ {x \in \mathbb{Z}\left. \right|x^2 \leq 9} \right\}\!, \\ C & = \left\{ {\left. {x \in \mathbb{Z}} \right|x \geq - 3} \right\}\!, & D & = \left\{ {1,2,3,4} \right\}\!\end{aligned}$$

. In each of the following, fill in the blank with one or more of the symbols $\subset$ , $\subseteq$ , $\not \subseteq$, $=$ , $\ne$, $\in$, $\text{ or } \notin$ so that the resulting statement is true. For each blank, include all symbols that result in a true statement. If none of these symbols makes a true statement, write nothing in the blank.

$\left\{ 3, 2, 1 \right\}$_____$D$

Assume that the universal set is the set of integers. Let

$$\begin{aligned}A & = \left\{ { - 3, - 2,2,3} \right\}\!, & B & = \left\{ {x \in \mathbb{Z}\left. \right|x^2 \leq 9} \right\}\!, \\ C & = \left\{ {\left. {x \in \mathbb{Z}} \right|x \geq - 3} \right\}\!, & D & = \left\{ {1,2,3,4} \right\}\!\end{aligned}$$

. In each of the following, fill in the blank with one or more of the symbols $\subset$ , $\subseteq$ , $\not \subseteq$, $=$ , $\ne$, $\in$, $\text{ or } \notin$ so that the resulting statement is true. For each blank, include all symbols that result in a true statement. If none of these symbols makes a true statement, write nothing in the blank.

$D$_____$\emptyset$

Assume that the universal set is the set of integers. Let

$$\begin{aligned}A & = \left\{ { - 3, - 2,2,3} \right\}\!, & B & = \left\{ {x \in \mathbb{Z}\left. \right|x^2 \leq 9} \right\}\!, \\ C & = \left\{ {\left. {x \in \mathbb{Z}} \right|x \geq - 3} \right\}\!, & D & = \left\{ {1,2,3,4} \right\}\!\end{aligned}$$

. In each of the following, fill in the blank with one or more of the symbols $\subset$ , $\subseteq$ , $\not \subseteq$, $=$ , $\ne$, $\in$, $\text{ or } \notin$ so that the resulting statement is true. For each blank, include all symbols that result in a true statement. If none of these symbols makes a true statement, write nothing in the blank.

$\mathbf{card}(A)$_____$\mathbf{card}(B)$

Assume that the universal set is the set of integers. Let

$$\begin{aligned}A & = \left\{ { - 3, - 2,2,3} \right\}\!, & B & = \left\{ {x \in \mathbb{Z}\left. \right|x^2 \leq 9} \right\}\!, \\ C & = \left\{ {\left. {x \in \mathbb{Z}} \right|x \geq - 3} \right\}\!, & D & = \left\{ {1,2,3,4} \right\}\!\end{aligned}$$

. In each of the following, fill in the blank with one or more of the symbols $\subset$ , $\subseteq$ , $\not \subseteq$, $=$ , $\ne$, $\in$, $\text{ or } \notin$ so that the resulting statement is true. For each blank, include all symbols that result in a true statement. If none of these symbols makes a true statement, write nothing in the blank.

$A$_____$\mathcal{P}(B)$

Write all of the proper subset relations that are possible using the sets of numbers $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, and $\mathbb{R}$.

For each statement, write a brief, clear explanation of why the statement is true or why it is false.

The set $\left\{ {a,b} \right\}$ is a subset of $\left\{ {a,c,d,e} \right\}$.

For each statement, write a brief, clear explanation of why the statement is true or why it is false.

The set $\left\{ { - 2,0,2} \right\}$ is equal to $\left\{ {x \in \mathbb{Z} \mid x\text{ is even and } x^2 \lt 5} \right\}$.

For each statement, write a brief, clear explanation of why the statement is true or why it is false.

The empty set $\emptyset$ is a subset of $\left\{ 1 \right\}$.

For each statement, write a brief, clear explanation of why the statement is true or why it is false.

If $A = \left\{ {a,b} \right\}$, then the set $\left\{ a \right\}$ is a subset of $\mathcal{P}( A )$.

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.

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