Decide whether the inferences are valid in each case

[nesan101]

I have two questions, it would be awesome if you could check my work / help. Thank you

Question 1.

. . .p -> r
. . .p v q
. . .not q
. . .therefore r

My work

. . .p - > r : Premise

. . .p v q : Premise

. . .q v p : (logically equivalent to 2)

. . .not q - > p : (logically equivalent to 3)

. . .not q : Premise

. . .therefore r : (hypothetical syllogism using 4 and 1)

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Question 2.

. . .\(\displaystyle \forall x \in R, p(x) \text{ v } q(x)\)

. . .\(\displaystyle \text {a} \in R\)

. . .\(\displaystyle q(a) - > r(a)\)

. . .\(\displaystyle \therefore p(a) \text { v } r(a)\)

My go at it:

. . .p(x) v q(x) : Premise

. . .p (a) v q(a) : Universal instantiation

. . .q(a) - > r(a) : Premise

. . .not q(a) v r(a) : Logically equivalent to above

Can I do?

. . .((a) v q(a)) ^ (not q(a) v r(a)) - > (p(a ) v r(a))

Is that wrong? If it's correct, I'm not sure why.

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Thank you.