Q 63. - Calculus

Question

Prove Theorem 12.10. That is, show that ${\left(s^e\right)}^e=s$ when S is a subset of $R^2$or $R^3$. 

Step-by-Step Solution

Answer

The theorem is proved "If $S$ is the subset of $R^2 or R^3$,then ${\left(s^e\right)}^e=s$

Step 1. Given information.

We have given subset ${\left(s^e\right)}^e=s$

Step 2: To find the complement of S

If S is the subset of $S^2$ then the complement of S is

$S^e=\left\{\left(x,y\right)\in R^2:x,y\notin s\right\}$

Now the complement of $S^e$ is :

${\left(s^e\right)}^e=\left\{\left(x,y\right)\in R^2:x,y\notin S^e\right\}$

According to the definition of complement of a set, it is clear that elements which are included in a set are not included in its complement. Similarly, the elements which are not included in the complement must be included in the original set. 

Thus, if $\left(x,y\right)\notin s^{e }$then $\left(x,y\right)\in s$

Therefore, 

$$\begin{gathered} {\left(s^e\right)}^e=\left\{\left(x,y\right)\in R^2:\left(x,y\right)\in s\right\} \\ {\left(s^e\right)}^e=s \end{gathered}$$