How to find a function to create a proper subset from intersections?

Question

So I'm currently trying to figure out how I might go about finding a function which, when evaluated on the intersection of two sets (Let's call them $X, Y$), then the following is true:

$$f(X\bigcap Y) \subset f(X) \bigcap f(Y)$$

What I'm struggling to do here is find a function $f$ which creates the required result of a proper subset; the values in $X,Y$ shouldn't really matter I don't think for this, but I'm very much stuck on finding an actual function (Such as like $f(x) = 2x$) for some sets $X = $ {1,2,3,4,5}, $Y =$ {2,3,8} (these are just dummy values; if picking specific values matters, I'd like to know why!) such that the function evaluated on $X\bigcap Y$ is again a proper subset of $f(X) \bigcap f(Y)$.

Can anyone help?

Answer 1

You may want to see something like this?

$X = [-1,0]$, $Y=[0,1] \Rightarrow X\cap Y = \{0\}$ and $f(X) = f(Y)=[0,1] $

In this case $f(X\cap Y) $ is a proper subset of $f(X)\cap f(Y)$

Answer 2

The simplest example is for $f$ to be a constant map and $X$,$Y$ disjoint.