Normally, I don't face difficulties in translating quantified statements in English to symbolic logic, but here I need advice. Thanks in advance!
Domain:
A set of employees who work at a company. Ingrid is one of the employees at the company.
- $S(x)$: $x$ was sick yesterday
- $W(x)$: $x$ went to work yesterday
- $V(x)$: $x$ went on vacation yesterday
1) Someone besides Ingrid was sick yesterday.
A. ∃x S(x) ∧ x ≠ Ingrid .....if this is correct then it doesn't talk about what if Ingrid was sick as there is a possibility of both he was sick or not.
2) Everyone besides Ingrid was sick yesterday.
A. ∀x S(x) ∧ x ≠ Ingrid .......same problem here also like mentioned for above one.
3) Someone who missed work was neither sick nor on vacation.
A. ∃x [ ¬W(x) → ¬ S(x) ∧ ¬ V(x) ] ....... Is it correct? I think it is.
1) Someone besides Ingrid was sick yesterday.
You are (mostly) correct, but you'll need to use parentheses to indicate the scope of the quantifier "$\exists$" in this case. $\exists x(\langle \text{whatever you need to say about} x\rangle).$
(The same is true when you are addressing anything about "all": $\forall x(\langle \text{whatever you need to say about} x \rangle).$ Remember to use parenthes!)
So, we simply modify your answer. Then we have $$\exists x (S(x) \land (x\neq \text{Ingrid}))\tag{1}$$
2) Everyone besides Ingrid was sick yesterday.
$$\forall x((x\neq \text{Ingrid})\rightarrow S(x))\tag{2}$$
3) Someone who missed work was neither sick nor on vacation.
$$∃x (\lnot W(x) \land ¬ S(x) \land ¬ V(x))\tag{3}$$
As Mauro indicated in the comments, when we use $\forall$, you can expect to use implication ($\rightarrow$). And when we use $\exists,$ you can expect to use conjuction ($\land$).