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Question 8 of 18

Refer to the following statement to answer parts (a) through (c) below:

You did not mow the lawn or you left the room a mess.

(a) Write the statement in symbolic form. Assign letters to simple statements that are not negated. Choose the correct answer:

A. Let [tex]p[/tex] = "you did not mow the lawn" and let [tex]q[/tex] = "you left the room a mess": [tex]\sim p \vee q[/tex]

B. Let [tex]p[/tex] = "you mowed the lawn" and let [tex]q[/tex] = "you left the room a mess": [tex]\sim p \wedge q[/tex]

C. Let [tex]p[/tex] = "you did not mow the lawn" and let [tex]q[/tex] = "you left the room a mess": [tex]\sim p \vee q[/tex]

D. Let [tex]p[/tex] = "you mowed the lawn" and let [tex]q[/tex] = "you left the room a mess": [tex]\sim p \vee q[/tex]

(b) Construct a truth table for the symbolic statement in part (a).

[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
$p$ & $q$ & $\sim p$ & $\sim p \vee q$ \\
\hline
T & T & F & T \\
\hline
T & F & F & F \\
\hline
F & T & T & T \\
\hline
F & F & T & T \\
\hline
\end{tabular}
\][/tex]

(c) Determine the truth value of the statement if you did not mow the lawn and you left the room a mess.

(Provide an explanation or choose the correct answer based on the truth table.)

Sagot :

GinnyAnsweridn
## a. Write the statement in symbolic form.
We are given the statement:

- "You did not mow the lawn or you left the room a mess."

Let's assign letters to the simple statements:
- Let [tex]\( p \)[/tex] be "You did not mow the lawn."
- Let [tex]\( q \)[/tex] be "You left the room a mess."

The statement to symbolize is:

- "You did not mow the lawn or you left the room a mess."

In symbolic form, this would be written as:
[tex]\[ \text{Answer: } \ p \vee q \][/tex]

### Choosing the correct answer:
A. [tex]\(\sim p \wedge q\)[/tex]: This represents "It is not the case that you did not mow the lawn AND you left the room a mess." -> Incorrect
B. [tex]\(\sim p \wedge q\)[/tex]: This represents "It is not the case that you mowed the lawn AND you left the room a mess." -> Incorrect
C. [tex]\( p \vee q \)[/tex]: This represents "You did not mow the lawn OR you left the room a mess." -> Correct
D. [tex]\(\sim p \vee q\)[/tex]: This represents "It is not the case that you mowed the lawn OR you left the room a mess." -> Incorrect

The correct answer is C.

## b. Construct a truth table for the symbolic statement in part (a).

Let's create the truth table for [tex]\( p \vee q \)[/tex]:

| [tex]\( p \)[/tex] | [tex]\( q \)[/tex] | [tex]\( p \)[/tex] | [tex]\( q \)[/tex] | [tex]\( \sim p \)[/tex] | [tex]\( \sim p \vee q \)[/tex] |
| ------ | ------ | ------ | ------ | ------ | ------ |
| T | T | F | T | F | T |
| T | F | F | F | T | F |
| F | T | T | T | F | T |
| F | F | T | F | T | T |
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