I need assistance with the following proofs. I am not sure how to prove that one set is contained in another set. Here are the things required to be proven:
Let $S$, $S_1$ and $S_2$ be non–empty sets in $R_n$. $ S^*$ = {$p|p^Tx \leq 0, \forall x \in S$}. Then
(a) $S \subseteq S^{**}$, where $S^{**}=(S^*)^*$
(b) $S_1 \subseteq S_2$ implies $S_2^* \subseteq S_1^*$
(c) $(K(S))^* = S^* $, $K(S)$ = { $ \sum x_i\lambda_i | x_i \in S, \lambda_i \geq 0, i=1,..., m ; m \in N $}
Hints: For (a) we have to prove, that any $x \in S$ is an element of $(S^*)^*$. To be an elment of $(S^*)^*$, for all $ p\in S^*$ it must hold that $x^t p \le 0$. But as $x^t p = p^t x$ and $x \in S$, $p\in S^*$, we have from the definition of $S^*$ that ...
(b) Let $p \in S_2^*$, we must show that $p \in S_1^*$, that is $p^t x \le 0$ for all $x \in S_1$, but as $p \in S_2^*$, we have $p^t x \le 0$ for all $x \in S_2$. Now use $S_1 \subseteq S_2$.
(c) For (c) note that $S \subseteq K(S)$, so (b) gives you one inclusion, for the other, let $p \in S^*$. We want to show that $p \in K(S)^*$, so let $x \in K(S)$, then $x = \sum_i \lambda_i x_i$, with $x_i \in S$ and $\lambda_i \ge 0$. Then $$ p^t x = \sum_i \lambda_i p^t x_i \le \cdots $$