Negation of "Either X is true, or Y is true, but not both" Negation of "Either X is true, or Y is true, but not both"
My attempt:
If seems that let X be true and Y be true, not X for X is false and not Y for Y is false. In order for the above statement to be True, we need: 
The negation of both X and Y to be true: negate(X and Y) -> not X or not Y
For "Either X is true, or Y is true, but not both" is equivalent to below:
((X or Y) and (not X or not Y))
The negation of the above (negation turns "and" into "or", and turns "or" into "and"):
((not X and not Y) or (X and Y))
Is this logical or? I am pretty lost...
 A: A basic approach is to see there only are 4 possibilities:


*

*$X$ and $Y$

*$X$ and not $Y$

*not $X$ and $Y$

*not $X$ and not $Y$
Your statement was (2) or (3), so its negation is (1) or (4).
A: I give a solution by formal calculation: translate the phrase into logical symbols, we have $\text{negation}\rightarrow\neg$, $\text{either X or Y is true}\rightarrow X\lor Y$, $\text{but not both}\rightarrow\land\neg(X\land Y)$. Then
$\neg((X\lor Y)\land\neg(X\land Y)=\neg(X\lor Y)\lor(\neg\neg(X\land Y))=(\neg X\land\neg Y)\lor(X\land Y)$
The final result is identical to yours.
A: Either $X$ is true, or $Y$  but not both is
($X$ OR $Y$) AND (NOT [$X$ AND $Y$])
Now the negation of $A$ AND $B$ is:
(not A) OR (not B).
So the negation is:
[NOT (X OR Y)] OR (NOT(NOT([X AND Y]))
And NOT(NOT(A)) is .. A so 
[NOT (X OR Y)] OR [X AND Y]
And the negation of A OR  B is:  (NOT A) AND (NOT B).
So 
(NOT X AND NOT Y) OR (X AND Y)
So the negation is
Either both X and Y are true, or both X and Y are false.
Maybe that is what it intuitively what you would have thought. 
