Consider the following propositions:

If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on.

If the sailing race is held, then the trophy will be awarded.

The trophy was not awarded.

(a) Write these propositions using the following propositional variables:

r : f : s : l : t :

It rains

It is foggy

The sailing race will be held

The lifesaving demonstration will go on The trophy will be awarded

(b) Using formal proof, show that the above propositions imply the conclusion It rained. Make sure to specify the rule you used in each line of your proof

2. (30 points) Consider the following propositions: "If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on." . "If the sailing race is held, then the trophy will be awarded." "The trophy was not awarded." (a) (10 points) Write these propositions using the following propositional variables: r : It rains : It is foggy S : The sailing race will be held 1 : The lifesaving demonstration will go on t : The trophy will be awarded (b) (20 points) Using formal proof, show that the above propositions imply the conclusion "It rained." Make sure to specify the rule you used in each line of your proof