Discrete Math Q1 - Q3 are word problems given as a sequence of hypotheses/ premi

Question

 Discrete Math  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.  2) 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.  3) 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.  For Q4, Q5 p,q,r,s,t,w are propositions,  ~ is the negation operator.  4) Show that the following argument is valid:     (~p  q) => r      s  ~q     ~t      p => t    (~p  r) => ~s  ------------------------   ~q  5) Show that the following argument is valid:       r => p     ~w  r     (t  q) => (p  s)      q  w     ~p     ~r => t  ------------------------    s 

Explanation / Answer

Please post one more question to get the remaining answer

1) The first statement is a valid arguement

Let going by bus or shuttle be p

Let going by taxi be t

and i will be broke be q

and i will be late for the appointment r

p -> r

t-> (not r and q)

not p -> t -> not r

Implies r, since if we didn't take shuttle or bus, then the only possible way to reach the appointment is taxi and in that case we will be broke

Hence the word problem is a valid arguement

2)

The statement is valid since

p-> represents the increase in interest rate then utility rate increases and housing market

q-> federal lones but not construction decrease

r-< increasing in interest and utility rate

p->q

q-<r

Hence the federal lones will become available

Therefore the given arguement is valid

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