A complement to the comments above.
It's worth mentioning that the meaning of the propositional connectives $\neg$, $\wedge$, $\vee$, $\to$ should not be regarded as a mere symbolic translation of the meaning of their English counterparts "not", "and", "or", "if ... then" respectively. See Hedman's A First Course in Logic (2004), p.1-2:
Unlike their English counterparts, these symbols represent concepts that are precise and invariable. The meaning of an English word, on the other hand, always depends on the context. For example, ∧ represents a concept that is similar but not identical to “and.” For atomic formulas A and B, A ∧ B always means the same as B∧A. This is not always true of the word “and.” The sentence
She became violently sick and she went to the doctor.
does not have the same meaning as
She went to the doctor and she became violently sick.
Likewise ∨ differs from “or.” Conversationally, the use of “A or B” often precludes
the possibility of both A and B. In propositional logic A∨B always means
either A or B or both A and B.
This is the case in the sentences above:
(1) Catch Billy a fish, and you will feed him for a day.
(2) Teach him to fish, and you'll feed him for life.
Note that the "and" here should not be interpreted as "$\wedge$". We have many similar cases:
(i) Jump and you die
We intuitively know that this sentence actually means:
(i') If you jump then you will die
Hence the argument is stated this way:
- $C \to D$
- $T \to L$
$\therefore \neg L \vee T$
Which is not valid (note that the conclusion says '$L \to T$').
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Since $p\lor\neg p\equiv\mathrm{T}$ and $q\lor\neg q\equiv\mathrm{T}$, you have
\begin{align*}
\mathrm{T}\lor\mathrm{T}&\equiv\mathrm{T}\\
&\equiv\mathrm{T}\land \mathrm{T}\\
&\equiv(p\lor\neg p)\land(q\lor\neg q).
\end{align*}
Best Answer
The last two should have the same answer, since the second one, $\lnot\forall x.P(x)$ says that it is not true that $P(x)$ holds for every $x$, while the third one, $\exists c.\lnot P(x)$ says that there exists an $x$ for which $P(x)$ does not hold. These mean exactly the same thing. "Not every crow is black" is the same as "There is a crow that is not black."
But your answer for the second one is not correct.
Mouse over for hint:
EDIT: Now your second one is correct, if you and I have the same idea for the part you indicated by "…", but it could be much simpler.
EDIT: Now your second one is incorrect again. $\lnot\forall x.P(x)$ says that $P(x)$ is not true for every $x$. It does $not$ say that $P(x)$ is true for any $x$; it might be false for all $x$.