Use rules of inference to show steps and reasons in the proof. If I take a bus o

Question

Use rules of inference to show steps and reasons in the proof. 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. If interest rates increase then utility rates increase and the housing market improves. Either federal loans become available or construction decreases. Interest rates increase and utility rates increase. Either construction grows or the housing market does not improve. Therefore, federal loans will become available. It is cold and not sunny today. We'll go for a run only if it is sunny. If we do not go for a run then it is not cold or we will go for a walk on the beach. We will watch a movie or we will not go for a walk on the beach. Therefore, we will watch a movie.

Explanation / Answer

1)

p = take bus
q = take subway
r = late for appointment
s = take taxi
t = I will be broke

p or q -> r           (i)
s -> ~r   and t           (ii)
~p and ~q -> s       (iii)
~r                   (iv)

Given ~r is true
Prove t is true

1. ~r                   premise (iv)
2. p or q -> r           premise (i)
3. ~(p or q)           Modus tollens (1,2)
4. ~p and ~q           DeMorgan (3)
5. ~p and ~q -> s       premise (iii)
6. s                   Modus ponens (4, 5)
7. s -> ~r and t       premise (ii)
8. ~r and t               Modus ponens (6,7)
9. t                   Decomposing a conjunction (8)

Therefore t is true
t = I will be broke

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2)

p = intereset rates increase
q = utility rates increase
r = housing market improves
s = federal loans become available
t = construction decreases

p -> q and r   (i)
s or t           (ii)
p and q           (iii)
~t or ~r       (iv)

prove s is true

1. p and q           premise (iii)
2. p               Decomposing a conjunction (1)
3. p -> q and r       premise (i)
4. q and r           Modus ponens (2, 3)
5. r               Decomposing a conjunction (4)
6. ~t or ~r           premise (iv)
7. ~t               Disjunctive syllogism (5, 6)
8. s or t           premise (ii)
9. s               Disjunctive syllogism (7, 8)

Therefore s is true
s = federal loans become available

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3)

p = It is cold
q = It is sunny day
r = we'll go for run
s = we'll go for a walk on the beach
t = we'll watch a movie

p and ~q       (i)
r -> q           (ii)
~r -> ~p or s   (iii)
t or ~s           (iv)

Prove, t is true

1. p and ~q           premise (i)
2. ~q               Decomposing a conjunction (1)
3. r -> q           premise (ii)
4. ~r               Modus tollens (3,4)
5. ~r -> ~p or s   premise (iii)
6. ~p or s           Modus ponens (5,6)
7. p               Decomposing a conjunction (1)
8. s               Disjunctive syllogism (6,7)
9. t or ~s           premise (iv)
10. t               Disjunctive syllogism (8,9)

Therefore, t is true
t = we'll watch a movie

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