Q 67. - Calculus

Question

Let S be a subset of $R^2$ or $R^3$. Prove that $\partial S=\partial \left(S^c\right)$

Step-by-Step Solution

Answer

The given statement is proved if S is the subset then$\partial S=\partial \left(S^c\right)$.

Step 1. Given information.

We have given S is a subset of $R^2 or R^3$.

Step 2: Prove the given statement.

Assume an element x'such that $x\in \partial S$. 

This means that the element 'x'is on the boundary of S. 

Every open disk D around this element 'x' will extend to both S and $S^c$.
Hence, 'x' should be included in the boundary of $S^c$ also.

Thus, $x\in \partial S^c$

Combine both the statements,

$\partial S=\partial \left(S^c\right)$