1. interior proofs

For sets A,B subsets of the Reals

a) int(A) intersect int(B) = int(A intersect B)
b) bd(A U B) is subset of bd(A) U bd(B)
c) Give examples of sets A and B where bd(A U B) = empty set and bd(A) = R = bd(B)

2. Originally Posted by Jdg6057
For sets A,B subsets of the Reals

a) int(A) intersect int(B) = int(A intersect B)
b) bd(A U B) is subset of bd(A) U bd(B)
c) Give examples of sets A and B where bd(A U B) = empty set and bd(A) = R = bd(B)
a) Let $x\in int(A)\cap int(B)$. Then $x\in int(A)$ and $x\in int(B)$. So $\exists~\epsilon_1,\epsilon_2$ such that $N(x,\epsilon_1)\subset A$ and $N(x,\epsilon_2)\subset B$. Taking $\epsilon=\min\{\epsilon_1,\epsilon_2\}$ guarantees that $N(x,\epsilon)\subset A\cap B$. So $x\in int(A\cap B)$ and $(int(A)\cap int(B))\subseteq int(A\cap B)$. See if you can prove the other direction.

b) Let $x\in bd(A\cup B)$. Then $\forall~\epsilon>0$, $N(x,\epsilon)\cap(A\cup B)^c\neq\emptyset$. Note that $(A\cup B)^c=A^c\cap B^c$.
Spoiler:
So $N(x,\epsilon)\cap A^c\cap B^c\neq\emptyset$. In particular, $N(x,\epsilon)\cap A^c\neq\emptyset$ and $N(x,\epsilon)\cap B^c\neq\emptyset$, so $x\in bd(A)\cup bd(B)$.

c) Let $X=\{z\in\mathbb{C}:\Im(z)\geq 0\}$ and $Y=\{z\in\mathbb{C}:\Im(z)\leq0\}$.

$bd(X)$ and $bd(Y)$ consist of all $z$ with $\Im(z)=0$; in other words, the real line. However, $X\cup Y=\mathbb{C}$, and $bd(\mathbb{C})=\emptyset$.