Does "either or " means XOR or OR? A set of r marbles is selected from an infinite supply of red, blue, white and yellow marbles. A selection must satisfy the condition that either the number of red marbles is even and the number of blue marbles is odd, or the number of white marbles is even and the number of yellow marbles is odd.
Does it mean a selection is invalid when both red is even and blue is odd as well as white is even and yellow is odd ? Do we discard this case while counting the number of selection ? 
 A: According to Cambridge Advanced Learner's Dictionary (CALD), it states that "either-or describes a situation in which there is a choice between two different plans of action, but both together are not possible". 
For example, it's an either-or situation - we can buy a new car this year or we can go on holiday, but we can't do both.
Therefore, either-or is XOR. 
BTW, in logic, "or" is often used to describe inclusive or, but in nature language, we also use it to describe exclusive-or situation. For example, "Would you like coffee or tea?" means either of them, but not both.   
A: In mathematical writing, "P or Q." always means "At least one of P,Q is true.", and indeed is the same as "$P \lor Q$" not "$P \oplus Q$" (P XOR Q). Actually, it can be argued that the English "or" is also inclusive when used as a boolean connective. I'm a native speaker and I have never seen any factual assertion "P or Q" in English where it is incorrect to interpret it as inclusive. (In "Do you want tea or coffee?" the "or" is not used as a boolean connective.)
