Why do we need logical quantifiers? I'm studying set theory, and sometimes my professor uses the quantifier $\forall$, and sometimes not. For example, the Axiom of Extensionality is stated as $\forall X,Y (X=Y \iff \forall z(z \in X \iff z \in Y))$
. Why do we need "$\forall z$"? I think $z \in X \iff z \in Y$ clearly makes sense and have the same meaning. In the class, we proved $U-(\bigcup_{i \in I} {A_i}) = \bigcap_{i \in I}{U-A_i}$ by
$$x \in (U - \bigcup_{i \in I}{}A_i) \iff x \in U \land \lnot(\exists i \in I (x \in A_i )) \iff x \in U \land \forall i \in I (x \notin A_i) \iff \forall i \in I (x \in U \land x \notin A_i) \iff x \in \bigcap _{i \in I}{U-A_i}$$
which did not use the quantifier.
 A: Omiting the leading universal quantifiers is a convention, but it can lead to confusion depending on the context. Say we have both forms of the following statement :


*

*$\forall x \ \forall y \ (x \in y \Longrightarrow y \notin x)$

*$x \in y \Longrightarrow y \notin x$
Imagine we're in a situation where this statement is needed. Also imagine that in this context we have a set that we named $x$ and another named $y$.
With form 1., no ambiguity. But with form 2., a machine would just see a very weak statement, mere folklore about our sets $x$ and $y$ (and nothing beyond that).
Now, if you suppress a non-leading quantifier, it's even worse. e.g. take the example you gave, the two statements have a totally different meaning :


*

*$\forall X,Y (X=Y \iff \forall z(z \in X \iff z \in Y))$

*$\forall X,Y (X=Y \iff (z \in X \iff z \in Y))$


*

*says the correct thing : "any two sets $X$ and $Y$ are equal iff they contain exactly the same elements"  

*takes a $z$ from the context and asserts : ""any two sets $X$ and $Y$ are equal iff whether one contains $z$, the other does, too."



All in all, don't be lazy, make the quantifiers explicit! ;)
A: Let X = {y,z}, Y = {z} and y /= z.
Even though z in X iff z in Y, does X = Y? 
A: *

*In words, the definition of set equality reads like this 



Whatever (set) $X$ and $Y$ may be, 
$(X = Y)$ iff [whatever (object) $x$ is a member of  $X$ is also a member of  $Y$, and
  reciprocally] 



*

*You need the $\forall$ quantifier to express these " whatever".


It can be recalled that this definion amounts to equating  set identity to reciprocal incluson, and you need a universal quantifier to express inclusion. 


*

*Also, you need the $\iff$ operator to express " reciprocally".

