# Math Help - interior points

1. ## interior points

hello! im having some trouble with this question which is due in tomorrow:

is it necessarily true that
int(AuB)=int(A) u int(B)?

im not sure how to construct a proof as i seem to keep just stating it with out proving it. Please help!

2. Let $A = \{ 0\leq x \leq 1 | x \in \mathbb{Q}\}$ and $B = \{ 0\leq x \leq 1 |x\not \in \mathbb{Q} \}$. Then $A \cup B = [0,1]$ and so $\mbox{int}(A\cup B) = (0,1)$ but $\mbox{int}(A) = \{ \}$ and $\mbox{int}(B) = \{ \}$.

3. how is int(AuB)= (0,1) surely int(AuB) would equal {} as well . i think i need to prove that this statement is true?

4. sorry i forgot to mention that A and B are subsets of R.(real numbers)

5. You have been shown that the proposition is false.
Here is a more mundane example.
$A = [0,1)\,\& \,B = [1,2)\quad \Rightarrow \quad A \cup B = [0,2)$
$Int(A) = (0,1)\,,\,Int(B) = (1,2)\,\& \,Int(A \cup B) = (0,2)$

BUT $(0,1) \cup (1,2) \ne (0,2)$