집합과 명제 전체 문제

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.

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

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 $\mathcal{P}( A )$.

Use the definitions of set intersection, set union, and set difference to write useful negations of these definitions. That is, complete each of the following sentences

$x \notin A \cap B$ if and only if $\ldots$.

Use the definitions of set intersection, set union, and set difference to write useful negations of these definitions. That is, complete each of the following sentences

$x \notin A \cup B$ if and only if $\ldots$.

Use the definitions of set intersection, set union, and set difference to write useful negations of these definitions. That is, complete each of the following sentences

$x \notin A - B$ if and only if $\ldots$.

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.

$A \cap B$

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.

$A \cup B$

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.

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

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.

$A^c \cap B^c$

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.

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