Please add me.
You should probably spend time learning what random variables are, from a formal math definition. This is what Bishop is using when he refers to X and Y in his sum and product rules.
A random variable is a function, with specific requirements. Which are the following, the domain of the function is an abstract sample space, think of all possible outcomes of an experiment. The codomain of the function is the real numbers. So a random variable maps an outcome(which is an element of the sample space) to a real number. What would the range of the function look like? All the set theory stuff follows when using random variables because functions can be constructed from naive set theory.
In his statement of the sum and product rules he is using a general description of having multiple random variables. Again all the set theory stuff is still applicable, you can just think of all possible 2-tuples when dealing with 2 random variables, since you will have a sample space for each variable.
A good exercise is to construct our own random variables. Say a coin and a 6 sided die. Setup 1: everything is indepedent, meaning the coin is flipped and the die is rolled seperately. Try to construct a function that maps all possible outcomes to their probabilities.
Setup 2: Have some dependence with die and coin. Say if you get heads you can only get even die outcomes, whatever you decide on, you need to construct the function so that it is valid.
Now try applying the sum and product rules to various subsets of the sample space(we call them events) of the functions you made, and it should work if you have done everything right :)