명제와 조건

Write the converse and contrapositive of each of the following conditional statements.

If it is not raining, then Laura is playing golf.

Write the converse and contrapositive of each of the following conditional statements.

If $a \ne b$, then $a^4 \ne b^4$.

Write the converse and contrapositive of each of the following conditional statements.

If $a$ is an odd integer, then $3a$ is an odd integer.

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 mow the lawn, then I will pay you $20.

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:

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

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.

Use the roster method to specify the truth set for each of the following open sentences. The universal set for each open sentence is the set of integers $\mathbb{Z}$.

$n + 7 = 4$.

Use the roster method to specify the truth set for each of the following open sentences. The universal set for each open sentence is the set of integers $\mathbb{Z}$.

$n^2 = 64$.

Use the roster method to specify the truth set for each of the following open sentences. The universal set for each open sentence is the set of integers $\mathbb{Z}$.

$\sqrt{n} \in \mathbb{N}$ and $n$ is less than 50.

Use the roster method to specify the truth set for each of the following open sentences. The universal set for each open sentence is the set of integers $\mathbb{Z}$.

$n$ is an odd integer that is greater than 2 and less than 14.