This is an abstract from a past exam paper at my university:
Below are my answers and reasoning. If you could tell me if my answers are incorrect or my reasoning is not rigorous enough, that would be helpful.
(i) False. A counterexample which disproves the proposition: $\neg p_{1}$, $p_{2}$ does not satisfy $p_{1}$ $\wedge$ $p_{2}$.
(ii) False. $(\neg q \rightarrow\neg p)$ $\iff$ $(p \rightarrow q)$, $(p \rightarrow q)$ $\not\iff$ $\neg (p \rightarrow q)$
(iii) True. The formula of any tautology will be such that any truth values assigned to its containing propositional atoms will always result in the compound proposition (in this case, 'the tautology') being true.
(iv) True. No truth values assigned to the formula's propositional atoms can output True. Thus this is a contradiction. Note that I used truth table to decide this, but will not iterate it here.
(v) True. If $x$ $<$ $y$, then $\exists j \in \mathbb{Z}$ : $x$ $<$ $\displaystyle\frac{(y-x)}{j}$ + $x$ $<$ $y$. $\space$ $\displaystyle\frac{(y-x)}{j}$ $\in \mathbb{Q}$.
Hence, $\exists z \in \mathbb{Q} (x < z < y)$.
(vi) False. There exists no integer larger than all integers.
Proof by Contradiction: Assume $\exists x \in \mathbb{Z} : \forall y \in \mathbb{Z}, x > y$. Then $\nexists z$ $\in$ $\mathbb{Z}$ $: z > x$.
But $\exists z \in \mathbb{Z} : z = x + 1$.
This is because $1$ $\in$ $\mathbb{Z}$ and the set $\mathbb{Z}$ is closed under addition.
Hence, $\exists z \in \mathbb{Z}$ $: z > x$ $\wedge$ $\nexists z$ $\in$ $\mathbb{Z}$ $: z > x$.
Hence, assuming $\exists x \in \mathbb{Z} : \forall y \in \mathbb{Z}, x > y$ leads us to a contradiction, thus $\exists x \in \mathbb{Z} : \forall y \in \mathbb{Z}, x > y$ is False.
(vii) True.
Direct Proof: Take any integer $y + 1$.
Thus, $\exists x \in \mathbb{Z} : x > y.$
Was this all OK?
(iii) is false. The only thing a tautology implies is another tautology! What is true is that any proposition implies a tautology, but that is not what the claim says.
Otherwise all your answers are correct, and you have good explanations for them!