Question

DISCRETE MATHEMATIC

Q1 - Q3 are word problems given as a sequence of hypotheses/
premises ending with "Therefore conclusion". Show that each
word problem is a valid argument.
Use rules of inference to show steps and reasons in the proof.

1) If I take a bus or subway then I'll be late for my
appointment.
If I take a taxi then I will be on time for my appointment and
I will be broke. If I don't take the subway and don't take a bus
then I will take a taxi. I will not be late for the appointment.
Therefore, I will be broke.

Answer 1

The conclusion "T" logically follows from the premises given and the argument is valid

Explanation:

Let us use notations to represent the steps

P: I take a bus

Q: I take the subway

R: I will be late for my appointment

S: I take a taxi

T: I will be broke

The given statement in symbolic form can be written as,

(P V Q) → R

S → (¬R ∧ T)

(¬Q ∧ ¬P) → S

¬R

___________________

∴ T

PROOF:

1. (¬Q ∧ ¬P) → S Premise

2. S → (¬R ∧ T) Premise

3. (¬Q ∧ ¬P) → (¬R ∧ T) (1), (2), Chain Rule

4. ¬(P ∨ Q) → (¬R ∧ T) (3), DeMorgan's law

5. (P ∨ Q) → R Premise

6. ¬R Premise

7. ¬(P ∨ Q) (5), (6), Modus Tollen's rule

8. ¬R ∧ T (4), (7), Modus Ponen's rule

9. T (8), Rule of Conjunction

Therefore the conclusion "T" logically follows from the given premises and the argument is valid.