MHB How Can Two Inequality Sets Be Combined into One?

[Barioth]

Hi everyone, let's stay I have two inequation set such as:

First one is A:=
$$X_1-X_2 \leq 1$$
$$X_1 \leq3$$
$$X_2 \geq 1$$
$$X_1,X_2 \geq 0$$

Second one is B:=
$$X_1+X_2 \geq 5$$
$$X_1\leq5$$
$$X_1\geq4$$
$$X_2\leq4$$
$$X_1,X_2 \geq 0$$

I had like to write it as a set $$C := A\oplus B$$, with C made of linear inequations too. I'm not so sure of how to tackle such problem, if anyone can help!

 

[Evgeny.Makarov]

re: Two inequation set into one.

What do you mean by $\oplus$?

 

[Barioth]

re: Two inequation set into one.

the XOR operation, sorry I should have said so!

 

[Evgeny.Makarov]

Re: Two inequation set into one.

The set of solutions to an inequality in two variables is a semi-plane. In particular, it is convex. Therefore, the set of solutions to several inequality is also convex as an intersection of convex sets. On the other hand, symmetric difference can act as set difference when one of the sets is inside another. Thus, it can turn two convex sets into a non-convex set. Therefore, the result is not always representable as the set of solutions of linear inequalities.