Intersection of a family of functions (counterexample) I always get a little bit confuse at this part. So i want to prove that:
$$\cap_{i\in I} f(A_i)\nsubseteq f(\cap_{i\in I} A_i)$$
So if i have $f(x)=x^2$,$A_1=\{1\}$ and $A_2=\{-1\}$. 
Since $A_1\cap A_2=\emptyset$ and $\cap_{í\in I}f(A_i)= \{1\}$.
Then $\{1\}\subseteq \emptyset$. But there's the contradiction. 
This counterexample helps me prove it?
 A: Your setup works.  How you want to phrase your solution really depends on how the question was given.

If you are asked to show that 
$$\bigcap_{i\in I} f(A_i) \subseteq f\left(\bigcap_{i\in I} A_i\right)$$
is not true in general, then you'll want to phrase your solution in terms of giving a counterexample.  So you could say, roughly,

For a counterexample, take $f(x) = x^2, A_1 = \{1\},$ and $A_2 =\{-1\}$.  Then $f(A_1) \cap f(A_2) = \{1\} \cap \{1\} = \{1\}$, and $f(A_1 \cap A_2) = f(\varnothing) = \varnothing$.  But $\{1\} \nsubseteq \varnothing$.  Therefore $\displaystyle \bigcap_{i\in I} f(A_i) \subseteq f\left(\bigcap_{i\in I} A_i\right)$ is not true in general.


If instead you are simply asked to show that 
$$\bigcap_{i\in I} f(A_i) \nsubseteq f\left(\bigcap_{i\in I} A_i\right)$$
for some $f$ and for some $\{A_i\}_{i\in I}$, then you're actually looking for an example, not a counterexample.  You want an example because you're looking for objects that satisfy the given statement.  In the first one we were looking for a counterexample because we wanted objects that failed to satisfy the given statement.  So in this case of looking for an example, you could say, roughly,

For an example, take $f(x) = x^2$, $A_1 = \{1\}$, and $A_2 = \{-1\}$.  Then $f(A_1) \cap f(A_2) = \{1\} \cap \{1\} = \{1\}$, and $f(A_1 \cap A_2) = f(\varnothing) = \varnothing$.  Since $\{1\} \nsubseteq \varnothing$, then $f(A_1) \cap f(A_2) \nsubseteq f(A_1 \cap A_2)$.  Therefore there is an $f$ and there is a family $\{A_i\}_{i\in I}$ such that $\displaystyle \bigcap_{i\in I} f(A_i) \nsubseteq f\left(\bigcap_{i\in I} A_i\right)$.


There is another case that's looks similar to but is very different from the second case.  If you were asked to show that
$$\bigcap_{i\in I} f(A_i) \nsubseteq f\left(\bigcap_{i\in I} A_i\right)$$
for all $f$ and all $\{A_i\}_{i\in I}$, then providing an example wouldn't suffice.  You'd have to prove it for all possible functions $f$ and all possible families of sets $\{A_i\}_{i\in I}$, so simply listing any amount of examples won't be enough.  I highly doubt this is what you were being asked to show, because this isn't actually true for all $f$ and all $\{A_i\}_{i\in I}$.
I mention this third case only because it's very important to be able to distinguish "Show this property is true for some object" (i.e., provide an example satisfying the property) and "Show this property is true for all objects" (i.e., provide a proof that all objects satisfy the property).
