If ralph doesnt do his homework or he doesnt feel sick, then he will go to the party and stay up late

By Direct Proof :

1.(∼H(x)∨∼S(x))→(P(x)∨L(x))1.(∼H(x)∨∼S(x))→(P(x)∨L(x))

2.P(x)→E(x)2.P(x)→E(x)

3.∼E(x)3.∼E(x)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

4.H(x)4.H(x)

5.P(x)→E(x)≡∼P(x)∨E(x)5.P(x)→E(x)≡∼P(x)∨E(x) by Material Implication

6.∼P(x)6.∼P(x) , #5 and #3 by Disjunctive Syllogism

7.∼P(x)∨∼L(x)7.∼P(x)∨∼L(x) , #6 by Addition ( I just add ∼∼L(x))

Since #7 is logically equivalent to ∼(P(x)∨L(x))∼(P(x)∨L(x)) by De Morgan's Law,

8.∼(∼H(x)∨∼S(x))8.∼(∼H(x)∨∼S(x)) , #1 and #7 by Modus Tollens.

Distributing the ∼∼, we'll have,

9.H(x)∧S(x)9.H(x)∧S(x) by De Morgan's and Double Negation

10.H(x)10.H(x) by Simplification ■

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answer for #1 is c

answer for #2 is d

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