Given following set:
$ X =\{x=(x_1, x_2, x_3, x_4)^T \in \{0, 1\}^4 : 83x_1 + 61x_2 + 49x_3 + 20x_4 \le 100 \} $
how I can check if:
$ P =\{x \in R^4 : 0 \le x \le (1,1,1,1)^T,$
$ 4x_1 + 3x_2 + 2x_3 + x_4 \le 4, $
$ x_1 + x_2 + x_3 \le 1, $
$ x_1 + x_4 \le 1 \} $
is a formulation for X and all vertex of P belong to X.
I'd like to know how can I proceed, after enumerate the points of $\{0, 1\}^4$ and check if belong to X and P:
x = [0 0 0 1] $\in X$ & $\in P$
x = [0 0 1 0] $\in X$ & $\in P$
x = [0 0 1 1] $\in X$ & $\in P$
x = [0 1 0 0] $\in X$ & $\in P$
x = [0 1 0 1] $\in X$ & $\in P$
x = [0 1 1 0] $\notin X$ & $\notin P$
x = [0 1 1 1] $\notin X$ & $\notin P$
x = [1 0 0 0] $\in X$ & $\in P$
x = [1 0 0 1] $\notin X$ & $\notin P$
x = [1 0 1 0] $\notin X$ & $\notin P$
x = [1 0 1 1] $\notin X$ & $\notin P$
x = [1 1 0 0] $\notin X$ & $\notin P$
x = [1 1 0 1] $\notin X$ & $\notin P$
x = [1 1 1 0] $\notin X$ & $\notin P$
x = [1 1 1 1] $\notin X$ & $\notin P$
x = [0 0 0 0] $\in X$ & $\in P$
I can not figure out, how to proceed, my idea is to find out all the vertex of P but I cannot figure how to proceed.
Futhermore if all vertex of P are integer and belong to X, can I assume that P is conv(X)?