Why does E neither equal to Y nor Z when B equals to Y? I was doing a quiz last night and one of the question is:
When B equals to Y, A equals to Z. When A does not equal to Z, E either equals to Y or Z, so:


*

*When B equals to Y, E equals to neither Y nor Z.

*When A equals to Z, Y or Z equals to E.

*When B does not equal to Y, E equals to neither Y nor Z.
The correct answer is 1.
This seems incorrect to me because:
a. "When A does not equal to Z, E either equals to Y or Z," this does not mean when A does equal to Z, E cannot to equal to Y or Z.
b. "When B equals to Y", "A does not equal to Z", and since this does not mean E cannot to equal to Y or Z, the statement "E equals to neither Y nor Z" is false.
 A: TLDR: All three options are incorrect

Let us refer to these statements by specific names.


*

*Statement $P$ will be "$B=Y$"

*Statement $Q$ will be "$A=Z$"

*Statement $R$ will be "$E=Y$ or $E=Z$"


We are told in the problem statement the following two things:
$$P\implies Q~~~~~\text{and}~~~~~ \neg Q\implies R$$
By looking at the contrapositives of these two statements, we also learn
$$\neg Q\implies \neg P~~~~~~\text{and}~~~~~~\neg R\implies Q$$
By combining these, we also learn that $\neg Q\implies R\wedge \neg P$, and by rearranging via contrapositive again, we have $P\vee \neg R\implies Q$ (which is technically a weaker statement than knowing both of the original two statements to each be separately true)

As for the available options in your multiple choice, they read in these terms as 
1) $P\implies \neg R$
2) $Q\implies R$
3) $\neg P\implies \neg R$

None of these three directly follow from the given hypotheses.
For 1) a counterexample would be when $P,Q$ and $R$ are all true, e.g. when $B=Y$ and $A=E=Z$.
For 2) a counterexample would be when $Q$ is true but $R$ is false, i.e. when $A=Z$ and when $E\neq Z$ and $E\neq Y$ ($P$ doesn't even matter in this one).
etc...
Any of these could be true, vacuously or otherwise, if we happened to restrict the universe in which we are considering (for example, if $A,B,E,Y,Z$ all are elements of $\Bbb F_2=\{0,1\}$ where we would get that statement $R$ happens to be a tautology and always be true in which case 2) would happen to be true)
