Predicate Logic and Logic Proofs(Review & Homework Questions)

Question

I'm working on some homework questions and I am struggling very hard with the logic proofs. I might have an incorrect answer for 1 of the predicate questions but I think my question makes some sort of sense. I think I have more of an issue with the logic proofs, any help or hints would be great. Thanks!

Logic Proof Questions

Decide whether the inferences are valid in each case. Show every step.

1)

(p V q) -> r

p

.'. r

2)

p -> r

p v q

~q

.'.r

1.

  (p v q) -> r

  ~(p v q) v r (using p - > q = ~p v q (logical equiv))

  ~(p) v r     (using p - > (p v q)  (Addition Inference Rule))  <- is this correct?

  This is where I got lost, I don't know how to get from that statement to r.

2.

  p -> r

  ~(p) v r   (using p - > q = ~p v q (logical equiv))

  (p -> q) ^ (q -> r)  (using ( (p -> q) ^ (q -> r) ) -> (p -> r) (hypothetical syllogism)

  (p -> q) ^ (~q v r)  ( using logic equiv)

  ~p v r (using modens tollens)

  This is where i got lost, also i think i used the rules wrong

Predicate Logic Questions

1) The equation x^2 + 2x + 1 = 0 has no solutions over the natural numbers.

2) Every positive real number has a unique positive real square root.

1. ( ∀x : | : (∄y : | : (x^2 + 2x + 1 = 0) = y) )

2. ( ∀x : ℝ | x > 0 : (Ǝy : ℝ | y > 0 : y^2 = x) ) (is this unique?)

I'd appreciate any help, I think i would need more help for the logic proof questions than the predicate questions because I am very lost on how to move from a statement to a different statement.

Answer 1

For 1) : $(p \lor q) \rightarrow r, p \vdash r$

we have :

1) $(p \lor q) \rightarrow r$ --- 1st premise

2) $p$ --- 2nd premise

3) $\lnot (p \lor q) \lor r$ --- from 1)

4) $p \lor q$ --- from 2) and the law : $\vdash p \rightarrow (p \lor q)$, by modus ponens

5) $r$ --- from 3) and 4) and disjunctive syllogism : from $\lnot A \lor B$ and $A$, infer : $B$.

For 2) : $p \rightarrow r, p \lor q, \lnot q \vdash r$

we have :

1) $p \rightarrow r$ --- 1st premise

2) $p \lor q$ --- 2nd premise

3) $\lnot q \rightarrow p$ --- from 2)

4) $\lnot q \rightarrow r$ --- from 3) and 1) by syllogism : from $A \rightarrow B$ and $B \rightarrow C$, infer : $A \rightarrow C$

5) $\lnot q$ --- 3rd premise

6) $r$ --- from 4) and 5) by modus ponens.

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For the predicate logic questions, I think that you have to translate them into formulae ...

If so, 1) must be :

$\lnot \exists n \in \mathbb N [(x+1)^2=0]$.

For 2:

$\forall x \in \mathbb R(x > 0 \rightarrow (\exists! y \in \mathbb R (y > 0 \land y^2 = x)))$.