Example of stronger programming formulation. So I was revising the course combinatorial optimization and I find the statement $$P_1 \text{ is a stronger formulation than } P_2 \text{ if } P_1 \subset P_2. $$ Can someone give me an example of two formulations for which one is stronger than the other? And why is it usefull?

Usually in combinatorial optimization, for example in integer linear programming, we want to relax the feasible set such that the problem become easier, such as relax it to a linear programming problem. 
The feasible set become larger that way. If the relaxed feasible set is too large, the approximation become crude. Hence we want the relaxation to be small. 
For example, you want to maximize $x$ such that $x \in P$ where $P=\{x: x<=11.5, x \in \mathbb{Z} \}$
Suppose we relax the integer criteria and solve the problem of maximize $x$ such that $x \in Q$, $Q=\{x: x<=11.5 \}$, then the optimal solution is $11.5$ for this relaxed problem. 
In contrast, if we further realize that there is no integer solution between $11$ and $11.5$ and solve the problem of maximize $x$ such that $x \in R$, $R=\{x: x<=11 \}$, then the optimal solution is $11$ for this relaxed problem. We can see that the solution is in fact closer to the original solution of the original problem, in fact they are equal.
