집합과 명제 전체 문제

If $P$ and $Q$ are statements, is the statement $\left( P \vee Q \right) \wedge \neg \left( P \wedge Q \right)$ logically equivalent to the statement $\left( P \wedge \neg Q \right) \vee \left( Q \wedge \neg P \right)$? Justify your conclusion.

Use previously proven logical equivalencies to prove each of the following logical equivalencies:

$\left[ {\neg P \to \left( {Q \wedge \neg Q} \right)} \right] \equiv P$

Use previously proven logical equivalencies to prove each of the following logical equivalencies:

$\left( P \leftrightarrow Q \right) \equiv \left( \neg P \vee Q \right) \wedge \left( \neg Q \vee P \right)$

Use previously proven logical equivalencies to prove each of the following logical equivalencies:

$\neg \left( P \leftrightarrow Q \right) \equiv \left( P \wedge \neg Q \right) \vee \left( Q \wedge \neg P \right)$

Use previously proven logical equivalencies to prove each of the following logical equivalencies:

$\left( P \to Q \right) \to R \equiv \left( P \wedge \neg Q \right) \vee R$

Use previously proven logical equivalencies to prove each of the following logical equivalencies:

$\left( P \to Q \right) \to R \equiv \left( \neg P \to R \right) \wedge \left( Q \to R \right)$

Use previously proven logical equivalencies to prove each of the following logical equivalencies:

$\left[ \left( P \wedge Q \right) \to \left( R \vee S \right) \right] \equiv \left[ \left( \neg R \wedge \neg S \right) \to \left( \neg P \vee \neg Q \right) \right]$

Use previously proven logical equivalencies to prove each of the following logical equivalencies:

$\left[ \left( P \wedge Q \right) \to \left( R \vee S \right) \right] \equiv \left[ \left( P \wedge Q \wedge \neg R \right) \to S \right]$

Use previously proven logical equivalencies to prove each of the following logical equivalencies:

$\left[ \left( P \wedge Q \right) \to \left( R \vee S \right) \right] \equiv \left( \neg P \vee \neg Q \vee R \vee S \right)$

Use previously proven logical equivalencies to prove each of the following logical equivalencies:

$\neg \left[ \left( P \wedge Q \right) \to \left( R \vee S \right) \right] \equiv \left( P \wedge Q \wedge \neg R \wedge \neg S \right)$

Let $A = \{ -4, -2, 0, 2, 4, 6, 8, \ldots \}$. Use correct set notation to indicate which of the following integers are in the set $A$ and which are not in the set $A$. For example, we could write $6 \in A$ and $5 \notin A$.

$$10 \qquad 22 \qquad 13 \qquad -3 \qquad 0 \qquad -12$$

Use correct set notation (using $=$ or $\subseteq$) to indicate which of the following sets are equal and which are subsets of one of the other sets.

$$\begin{aligned}A & = \{3, 6, 9 \} & B & = \{6, 9, 3, 6 \} \\ C & = \{3, 6, 9, \ldots \} & D & = \{ 3, 6, 7, 9 \} \\ E & = \{9, 12, 15, \ldots \} & F & = \{9, 7, 6, 2 \}\end{aligned}$$

Assume the universal set for all variables is $\mathbb{Z}$ and let $P( x )$ be the predicate “$x^2 \leq 4$.”

(i) Find two values of $x$ for which $P( x )$ is false.

(ii) Find two values of $x$ for which $P( x )$ is true.

(iii) Use the roster method to specify the set of all $x$ for which $P( x )$ is true.

Assume the universal set for all variables is $\mathbb{Z}$, and let $R( {x,\;y,\;z} )$ be the predicate “$x^2 + y^2 = z^2$.”

(i) Find two different examples for which $R( {x, y, z})$ is false.

(ii) Find two different examples for which $R( {x, y, z})$ is true.

Let $P( x )$ be the predicate “$x^2 \leq 9$.”

If the universal set is $\mathbb{R}$, describe the truth set of $P(x)$ using English and write the truth set of $P( x )$ using set builder notation.

Let $P( x )$ be the predicate “$x^2 \leq 9$.”

If the universal set is $\mathbb{Z}$, then what is the truth set of $P( x )$? Describe this set using English and then use the roster method to specify all the elements of this truth set.

Each of the following sets is defined using the roster method.

$$\begin{aligned}A & = \left\{1, 5, 9, 13, \ldots \right\} \\ B & = \left\{\ldots, -8, -6, -4, -2 , 0 \right\} \\ C & = \left\{\sqrt{2}, \left( \sqrt{2}\right)^3, \left( \sqrt{2}\right)^5, \ldots \,\right\} \\ D & = \left\{1, 3, 9, 27, \ldots \right\}\end{aligned}$$

Determine four elements of each set other than the ones listed using the roster method.

Each of the following sets is defined using the roster method.

$$\begin{aligned}A & = \left\{1, 5, 9, 13, \ldots \right\} \\ B & = \left\{\ldots, -8, -6, -4, -2 , 0 \right\} \\ C & = \left\{\sqrt{2}, \left( \sqrt{2}\right)^3, \left( \sqrt{2}\right)^5, \ldots \,\right\} \\ D & = \left\{1, 3, 9, 27, \ldots \right\}\end{aligned}$$

Use set builder notation to describe each set.

Use the roster method to specify the elements in each of the following sets and then write a sentence in English describing the set.

$\left\{ x \in \mathbb{R} \mid 2x^2 + 3x - 2 = 0 \right\}$

Use the roster method to specify the elements in each of the following sets and then write a sentence in English describing the set.

$\left\{ {x \in \mathbb{Z}} \mid 2x^2 + 3x - 2 = 0 \right\}$