Predicate Logic Proof Question

Question

I am struggling really hard with proofs I cannot seem to understand them at all no matter how hard i try. I'm thinking of getting a tutor because questions like this I just give up and fail on. Any hints or help would be appreciate, thanks.

Question:

Decide whether the inferences are valid in each case. Give the reason behind each step. Do not use truth tables in this question.

∀x ∈ ℝ, p(x) v q(x)

a ∈ ℝ

q(a) -> r(a)

.'. p(a) v r(a)

My Solution which is incorrect >

∀x ∈ ℝ, p(x) v q(x) (premise)

a ∈ ℝ (premise)

p(a) v q(a) (universal instantiation from (2))

p(a) (using simplification (1))

q(a) (using simplification (2))

I don't know this line(how do i find r to get to the next step?)

q(a) -> r(a)

¬q(a) v r(a) (logical equiv (7))

.'. p(a) v r(a)

Answer 1

It is quite similar to your previous post, case 2 : $p \rightarrow r, p \lor q, \lnot q \vdash r$.

Now we have :

$p(a) \lor q(a), q(a) \rightarrow r(a) \vdash p(a) \lor r(a)$

1) $p \lor q$ --- 1st premise

2) $q \rightarrow r$ --- 2nd premise

3) $\lnot p \rightarrow q$ --- from 1)

4) $\lnot p \rightarrow r$ --- from 3) and 2) by syllogism : from $A \rightarrow B$ and $B \rightarrow C$, infer : $A \rightarrow C$

5) $p \lor r$ --- from 4) .