So the initial task is to show that $X, Y$ and $XY$ are pairwise independent but not independent given that $X,Y$ are two independent r.v. on some probability space such that $P_X = P_Y = \frac{1}{2}(\delta_1 + \delta_{-1})$.
I've firstly tried to write down the cdf in a more intuitive way
$$F_X(x) = \begin{cases} 0,x <-1 \\ \frac{1}{2}, x \in [-1,1) \\ 1, x\ge1 \end{cases} $$
and the corresponding mass function:
$$f_X(x) = \begin{cases} 0,x \ne \pm1 \\ \frac{1}{2}, else \end{cases} $$
and the same for $Y$
But where to go from here... I realize that we need to construct the same functions for $XY$ and then to work with them but I have no idea how to proceed.