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Two inequalities set into one.

Barioth

Member
Jan 17, 2013
52
Hi everyone, let's stay I have two inequation set such as:

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

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

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

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
re: Two inequation set into one.

What do you mean by $\oplus$?
 

Barioth

Member
Jan 17, 2013
52
re: Two inequation set into one.

the XOR operation, sorry I should have said so!
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
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.