The easiest way to do this is if you have some function of two random variables (no matter what it is), if the probability P(X=x,Y=y) = P(X=x)P(Y=y) then you have independence. If the PDF is separable, then you have independence.
Also if you can argue in an algebraic or other way that P(X|Y) = P(X) and P(Y|X) = P(Y) then you get the same end result. In other words, if knowing Y doesn't tell you anything about X and vice-versa, then you have done the exact same thing.
Another way of thinking about this is to show that for any function Y = f(X), no function exists to relate the two variables together.
All statements are all equivalent.