집합과 명제 전체 문제

For statements $P$ and $Q$:

Use a truth table to show that $\left( {P \vee \neg P} \right)$ is a tautology.

For statements $P$ and $Q$:

Use a truth table to show that $\left( {P \wedge \neg P} \right)$ is a contradiction.

For statements $P$, $Q$, and $R$:

Show that $\left[ (P \to Q) \wedge P \right] \to Q$ is a tautology.

Note: In symbolic logic, this is an important logical argument form called modus ponens.

For statements $P$, $Q$, and $R$:

Show that $\left[ (P \to Q) \wedge (Q \to R) \right] \to (P \to R)$ is a tautology.

Note: In symbolic logic, this is an important logical argument form called syllogism.

Use previously proven logical equivalencies to prove each of the following logical equivalencies about conditionals with conjunctions:

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

Use previously proven logical equivalencies to prove each of the following logical equivalencies about conditionals with conjunctions:

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

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:

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

For each of the following, use a counterexample to show that the statement is false. Then write the negation of the statement in English, without using symbols for quantifiers.

$\left( {\forall m \in \mathbb{Z}} \right)\left( {m^2 \text{ is even } } \right)$.

For each of the following, use a counterexample to show that the statement is false. Then write the negation of the statement in English, without using symbols for quantifiers.

$\left( {\forall x \in \mathbb{R}} \right)\left( {x^2 > 0} \right)$.

For each of the following, use a counterexample to show that the statement is false. Then write the negation of the statement in English, without using symbols for quantifiers.

For each real number $x$, $\sqrt{x} \in \mathbb{R}$.

For each of the following, use a counterexample to show that the statement is false. Then write the negation of the statement in English, without using symbols for quantifiers.

$\left( \forall m \in \mathbb{Z} \right) \left( \dfrac{m}{3} \in \mathbb{Z} \right)$.