Question

\[x \in A\ or \ x \in B\] If I assume the former then, \[x \notin B\]?

Answer 1

nope. A may be in A and B.
http://en.wikipedia.org/wiki/Union_%28set_theory%29

Answer 2

A and B means the intersection of A and B

Answer 3

oops I meant x may be in A and B

Answer 4

Exactly. Suppose A = { 1, 2 }, B = { 2, 3 }
then it is true that:
2 in A or 2 in B

and in fact both are true.