집합의 뜻과 포함 관계

Let the universal set be $U = \left\{ {1,2,3,4,5,6} \right\}$, and let

$$A = \left\{ {1,2,4} \right\}\!, B = \left\{ {1,2,3,5} \right\}\!, C = \left\{ {\left. {x \in U} \right|x^2 \leq 2} \right\}\!$$

.

In each of the following, fill in the blank with one or more of the symbols $\subset, \\ \subseteq, =, \ne, \in ,\text{ or } \notin$ 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.

$\left\{ {1, 2} \right\}$_____$C$

Let the universal set be $U = \left\{ {1,2,3,4,5,6} \right\}$, and let

$$A = \left\{ {1,2,4} \right\}\!, B = \left\{ {1,2,3,5} \right\}\!, C = \left\{ {\left. {x \in U} \right|x^2 \leq 2} \right\}\!$$

.

In each of the following, fill in the blank with one or more of the symbols $\subset, \\ \subseteq, =, \ne, \in ,\text{ or } \notin$ 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.

$\left\{ {4, 2, 1} \right\}$_____$A$

Let the universal set be $U = \left\{ {1,2,3,4,5,6} \right\}$, and let

$$A = \left\{ {1,2,4} \right\}\!, B = \left\{ {1,2,3,5} \right\}\!, C = \left\{ {\left. {x \in U} \right|x^2 \leq 2} \right\}\!$$

.

In each of the following, fill in the blank with one or more of the symbols $\subset, \\ \subseteq, =, \ne, \in ,\text{ or } \notin$ 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.

$B$_____$\emptyset$

Assume the universal set is the set of real numbers. Let

$$\begin{aligned}A & = \left\{ { - 3, - 2,2,3} \right\}\!, & B & = \left\{ {\left. {x \in \mathbb{R}} \right|x^2 = 4\text{ or } x^2 = 9} \right\}\!, \\ C & = \left\{ {x \in \mathbb{R}\left. \right|x^2 + 2 = 0} \right\}\!, & D & = \left\{ {x \in \mathbb{R}\left. \right|x > 0} \right\}\!\end{aligned}$$

. Respond to each of the following questions. In each case, explain your answer.

Is the set $A$ equal to the set $B$?

Assume the universal set is the set of real numbers. Let

$$\begin{aligned}A & = \left\{ { - 3, - 2,2,3} \right\}\!, & B & = \left\{ {\left. {x \in \mathbb{R}} \right|x^2 = 4\text{ or } x^2 = 9} \right\}\!, \\ C & = \left\{ {x \in \mathbb{R}\left. \right|x^2 + 2 = 0} \right\}\!, & D & = \left\{ {x \in \mathbb{R}\left. \right|x > 0} \right\}\!\end{aligned}$$

. Respond to each of the following questions. In each case, explain your answer.

Is the set $A$ a subset of the set $B$?

Assume the universal set is the set of real numbers. Let

$$\begin{aligned}A & = \left\{ { - 3, - 2,2,3} \right\}\!, & B & = \left\{ {\left. {x \in \mathbb{R}} \right|x^2 = 4\text{ or } x^2 = 9} \right\}\!, \\ C & = \left\{ {x \in \mathbb{R}\left. \right|x^2 + 2 = 0} \right\}\!, & D & = \left\{ {x \in \mathbb{R}\left. \right|x > 0} \right\}\!\end{aligned}$$

. Respond to each of the following questions. In each case, explain your answer.

Is the set $C$ equal to the set $D$?

Assume the universal set is the set of real numbers. Let

$$\begin{aligned}A & = \left\{ { - 3, - 2,2,3} \right\}\!, & B & = \left\{ {\left. {x \in \mathbb{R}} \right|x^2 = 4\text{ or } x^2 = 9} \right\}\!, \\ C & = \left\{ {x \in \mathbb{R}\left. \right|x^2 + 2 = 0} \right\}\!, & D & = \left\{ {x \in \mathbb{R}\left. \right|x > 0} \right\}\!\end{aligned}$$

. Respond to each of the following questions. In each case, explain your answer.

Is the set $C$ a subset of the set $D$?

Assume the universal set is the set of real numbers. Let

$$\begin{aligned}A & = \left\{ { - 3, - 2,2,3} \right\}\!, & B & = \left\{ {\left. {x \in \mathbb{R}} \right|x^2 = 4\text{ or } x^2 = 9} \right\}\!, \\ C & = \left\{ {x \in \mathbb{R}\left. \right|x^2 + 2 = 0} \right\}\!, & D & = \left\{ {x \in \mathbb{R}\left. \right|x > 0} \right\}\!\end{aligned}$$

. Respond to each of the following questions. In each case, explain your answer.

Is the set $A$ a subset of the set $D$?

Explain why

the set $\left\{ {a,b} \right\}$ is equal to the set $\left\{ {b,a} \right\}$.

Explain why

the set $\left\{ {a,b,b,a,c} \right\}$ is equal to the set $\left\{ {b,c,a} \right\}$.

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.

$A$_____$B$

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.

5_____$C$

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.

$A$_____$C$

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.

$\left\{ 1, 2 \right\}$_____$A$

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.

4_____$B$

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.

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

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.

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

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.

$\emptyset$_____$A$

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.

$\left\{ 5 \right\}$_____$C$

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.

$\left\{ 1, 2 \right\}$_____$B$