Some definitions:
Let X be a non-empty subset of \mathbb{R}^n (for simplicities sake) . Then x \in X is called an interior point of X if there exists \epsilon > 0 such that the open ball centered at x, radius \epsilon is entirely contained in X. i.e B(x,\epsilon) \subseteq X.
We define the interior of A as such: int A = \left\{ x \in X : x \mbox{ is an interior point of X } \right\}.
We define the closure as such: \overline{A} = \left\{ x \in \mathbb{R}^n : \forall \epsilon > 0 , B(x,\epsilon) \cap X \neq \varnothing \right\}.
The Boundary of X is defined as such: \partial X = \overline{A} \backslash int A
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Now that the definitions are clear, finding the boundary of a set is almost algorithmic.
Example: X = (0,1). Every point in X is clearly an interior point. However you want to see it (by the definition, characterization as the set of all limits of convergent sequences in X, whatever you like), its closure is \overline{X} = [0,1]. Then \partial X = [0,1] \backslash (0,1) = \{ 0, 1 \}
"True or false:
Let S be any set in \mathbb{R}^2. The boundary of S is the set of points contained in S which are not in the interior of S."
Take a look at the example. It should give you the idea.
A general hint: Don't build intuition too "far away" from the defined terms, gain your intuition from the definitions. When you are too far away, you may think of terms in a sense that they are not defined as (in this case the question is trying to do that with "Boundary"). It
s better if you mix up having intuition for certain concepts (which sets are open, what is their closure), and then apply definitions to find the boundary.
