Set operations on connected sets in $R^2$ An exercise wants me to give an example of the following in $R^2$


*

*$A$ and $B$ are connected but $A \cap B$ is not.

*$A$ and $B$ are connected but $A \setminus B$ is not.

*$A$ and $B$ are not connected but $A \cup B$ is.


I think I found an example for 3. Take two curves with "holes" in them but holes should not coincide. Then their unions will fill their respective "holes" and will be connected. As you can see, I am moving only by geometric intuition. But I think this is the purpose of this kind of exercise.
So, what are some examples of 1 and 2, and also if you have, better examples of 3.
 A: *

*$A=$ line segment from $(-1,0)$ to $(1,0)$; $B=$ circle centered at $(0,0)$ with radius $1$.

*$A=$ line segment from $(-1,0)$ to $(1,0)$; $B=$ line segment from $\left(-\frac12,0\right)$ to $\left(\frac12,0\right)$.

A: For (1), picture a "peanut-shaped" curve, with two buges centered on about $(0,1$) and $(0,-1)$ and an identical curve displaced horizontally such that the two curves kiss tangentially at two points.  Let $A$ be the interior plus boundary of one curve, $B$ of the other.
For (2), take an ellipse and a circle centered on the intersectoin of its major and minor axes, such that the circle is tangent to the ellipse at two points.
A: 
$1)$  Take two circles $A,B$ with same radii and different centers and intersect them using a translation.The intersection in of these circles woulb be a two point set namely $A=\{(a_1,a_2),(b_1.b_2)\}$.Then $A$ is not a connected set as a union of two singletons which are closed sets with respect to the usual topology  on the plane.

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$2)$Take a closed disc $B(x_0,r)$as $A$ and a line segment $B$ which passes from its center and connects two antipodal points.These two sets are connected and $B(x_0,r)$ \ $B$ is not connected.

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$3)$ Take a  circle $A$ which misses two antipodal points and $B$ the set which contains those two points..clearly these two sets are disconected but their union is the circle which connected.

I hope this helps.
