Answer :
## 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 |
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 |
