Why is $A ∪ B$ called "A or B" when $A ∪ B$ means all elements in $A$, $B$ and their intersection? Definition: $A ∪ B$ is the set of elements that is contained in either $A$ or $B$, or in both.
My question: why is $A ∪ B$ called "A or B" when $A ∪ B$ is literally everything that is in $A$,$B$ and their intersection.
For example, a Venn diagram of $A ∪ B$ is having both circles A and B shaded in: $A$, $B$, and  $A ∩ B$. However, my confusion is that I interpret the definition above as selecting only one of the following, since it says "or":

*

*$A = A ∩ B^{c}$


*$B = A^{c} ∩ B$


*"in both" = $A ∩ B$
It is clear that if I combine all three subsets above, then I will have $A ∪ B$. But the definition says "or" which makes me think I am selecting strictly one of the three possible subsets above. So why is $A ∪ B$ called "A or B" rather than "A,B, A and B"
 A: That's because, in logic terms, it means "$A$ or $B$".
Imagine you have the universe $U$ that contains the sets $A$ and $B$, and imagine you are going to choose a random point of the universe $U$. After you have chosen it, you will obtain a TRUE result if that point verifies that it's contained in $A$. Also, if that point is contained in $B$, the result will also be TRUE, and a FALSE result if none of these conditions are verified. Obviously, when choosing a point that belongs both to $A$ and $B$, the outcome will be true because it verifies it's in $A$. Then, it's clear that the set that will lead you to TRUE results is $A\cup B$, to be said, "$A$ or $B$" not excluding intersection. The other possible definitions you're trying to give to it are assuming that your "OR" condition is exclusive, but the operator does not mean that.
To be said, the logical operator $\cup$ equals the operator "OR" and that's not excluding the intersection. If you wanted to exclude the intersection, $(A\setminus B)\cup(B\setminus A)$ is the correct expression, since $\cup$ will include any point that belongs to $A$ or $B$ not caring about intersection (a point that belongs to $A\cap B$ belongs to $A$, so it must be inside $A\cup B$).
A: Well, two statements, first statement ($p$) "$x$ in $A$", second statement ($q$) $x$ in $B$. $p$ or $q$ true iff at least one of them true.
