명제와 조건

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

I will wash the car or I will mow the lawn.

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

If you graduate from college, then you will get a job or you will go to graduate school.

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

If I play tennis, then I will wash the car or I will do the dishes.

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

If you clean your room or do the dishes, then you can go to see a movie.

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

It is warm outside and if it does not rain, then I will play golf.

Use truth tables to establish each of the following logical equivalencies dealing with biconditional statements:

$\left( {P \leftrightarrow Q} \right) \equiv \left( {P \to Q} \right) \wedge \left( {Q \to P} \right)$

Use truth tables to establish each of the following logical equivalencies dealing with biconditional statements:

$\left( {P \leftrightarrow Q} \right) \equiv \left( {Q \leftrightarrow P} \right)$

Use truth tables to establish each of the following logical equivalencies dealing with biconditional statements:

$\left( {P \leftrightarrow Q} \right) \equiv \left( {\neg P \leftrightarrow \neg Q} \right)$

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:

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