QUESTION IMAGE
Question
refer to the following statement to answer parts (a) through (c) below. i dress quickly or take the express train, and i do not get to work on time.
a. write the statement in symbolic form. assign letters to simple statements that are not negated. choose the correct answer below.
a. let p = i dress quickly, let q = i take the express train, and let r = i get to work on time; (p ∨ q) ∧ ¬r
b. let p = i dress quickly, let q = i take the express train, and let r = i get to work on time; (p ∨ q) ∨ ¬r
c. let p = i dress quickly, let q = i take the express train, and let r = i do not get to work on time; (p ∧ q) ∨ ¬r
d. let p = i dress quickly, let q = i take the express train, and let r = i do not get to work on time; (p ∨ q) ∧ ¬r
b. construct a truth table for the symbolic statement in part (a).
| p | q | r | p ∨ q | ¬r | (p ∨ q) ∧ ¬r |
|---|
Part (a)
Step 1: Identify Simple Statements
Let \( p \) = "I dress quickly", \( q \) = "I take the express train", \( r \) = "I get to work on time". The negation of \( r \) (denoted \(
eg r \)) is "I do not get to work on time". The original statement is "I dress quickly or take the express train, and I do not get to work on time", which is \( (p \vee q) \wedge
eg r \). Now check the options:
- Option A: \( p \) = "I dress quickly", \( q \) = "I take the express train", \( r \) = "I get to work on time", formula \( (p \vee q) \wedge
eg r \) – matches.
- Option B: Formula is \( (p \vee q) \vee
eg r \) – incorrect (uses "or" instead of "and").
- Option C: Formula is \( (p \wedge q) \vee
eg r \) – incorrect (uses "and" for \( p,q \) instead of "or").
- Option D: Formula is \( (p \vee q) \wedge
eg r \) but \( r \) is "I do not get to work on time" (wrong, \( r \) should be "I get to work on time" for \(
eg r \) to be correct negation) – incorrect.
Step 2: Confirm the Correct Option
Only Option A has the correct simple statements and logical connectives.
Step 1: List All Truth Values for \( p, q, r \)
There are \( 2^3 = 8 \) combinations for \( p, q, r \) (each can be T or F):
| \( p \) | \( q \) | \( r \) |
|---|---|---|
| T | T | F |
| T | F | T |
| T | F | F |
| F | T | T |
| F | T | F |
| F | F | T |
| F | F | F |
Step 2: Compute \( p \vee q \)
\( p \vee q \) is T if \( p \) or \( q \) is T:
| \( p \) | \( q \) | \( r \) | \( p \vee q \) |
|---|---|---|---|
| T | T | F | T |
| T | F | T | T |
| T | F | F | T |
| F | T | T | T |
| F | T | F | T |
| F | F | T | F |
| F | F | F | F |
Step 3: Compute \(
eg r \)
\(
eg r \) is T when \( r \) is F, F when \( r \) is T:
| \( p \) | \( q \) | \( r \) | \( p \vee q \) | \( |
eg r \) |
| T | T | T | T | F |
| T | T | F | T | T |
| T | F | T | T | F |
| T | F | F | T | T |
| F | T | T | T | F |
| F | T | F | T | T |
| F | F | T | F | F |
| F | F | F | F | T |
Step 4: Compute \( (p \vee q) \wedge
eg r \)
This is T only when both \( p \vee q \) and \(
eg r \) are T:
| \( p \) | \( q \) | \( r \) | \( p \vee q \) | \( |
eg r \) | \( (p \vee q) \wedge
eg r \) |
| T | T | T | T | F | F |
| T | T | F | T | T | T |
| T | F | T | T | F | F |
| T | F | F | T | T | T |
| F | T | T | T | F | F |
| F | T | F | T | T | T |
| F | F | T | F | F | F |
| F | F | F | F | T | F |
Snap & solve any problem in the app
Get step-by-step solutions on Sovi AI
Photo-based solutions with guided steps
Explore more problems and detailed explanations
A. let \( p \) = I dress quickly, let \( q \) = I take the express train, and let \( r \) = I get to work on time; \( (p \vee q) \wedge
eg r \)