Let $$\max\left\{c^T \cdot x \mid A \cdot x \leq b, x \geq 0\right\}$$ be an arbitrary linear program and let $M$ be its solution set. Is $M$ always bounded?
I think the solution set of linear programming problem is not always bounded because let's say the linear programming problem is infeasible, so then $M$ is empty. Is that enough of reasoning? It can also be unbounded I think if the value of its objective function can be made arbitrarily large so in that case it would be unbounded since its optimum value is $+ \infty$. And if this is not the case then the LP is bounded but as we see this is not always the case. So the statement is false, right?
If $M$ is empty, it is bounded.
Also note that, $M$ is a solution set, not a linear programming problem.
The statement is indeed false, here is a way to make $M$ unbounded. We let $A=0$ and $x=0$ and $c=0$. Hence $M=\{ x \in \mathbb{R}^n|x \ge 0\}$. It is unbounded. To see it clearly note that $ke \in M$, where $k \ge 0$ can be made arbitrarily large and $e$ is the all ones vector.