Prove that each argument is valid by replacing each proposition with a variable

Question

Prove that each argument is valid by replacing each proposition with a variable to obtain the form of the argument. Then use the rules of inference to prove that the form is valid.
a)
If I drive on the freeway, I will see the fire.
I will drive on the freeway or take surface streets (or both).
I am not going to take surface streets.
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I will see the fire.

(b)
If it was not foggy or it didn't rain (or both), then the race was held and there was a trophy ceremony.
The trophy ceremony was not held.
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It rained.

(c)
If I work out hard, then I am sore.
If I am sore, I take an aspirin.
I did not take an aspirin.
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I did not work out hard.

Explanation / Answer

a) D - Drive the freeway
   F - See the fire
   S - Drive the surface street

   first statement D->F
   Second statement - D, S , D or S
   Third statement - not S  

   Given the premise, not S implies D and D imples F. Hence the conclusion is valid.

b) F - it is foggy
   R - it rained
   A - race will held
   T - Troph ceremony held

   first statement - not F and not R --->A ---> T
   Second statement - not T

   not T implies not A and not A means either F or R. So we can assume R. So the conclusion is ok.

c) H - work hards
   S - i am sore
   A - take asprin

   first statement - H ->S
   Second statement - S->A
   third statement - not A

   not A implies not S and not S imples not H.So the conclusion holds good.

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