How Can Two Inequality Sets Be Combined into One?

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    Inequalities Set
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Discussion Overview

The discussion revolves around the combination of two sets of inequalities, referred to as sets A and B, into a new set C using the XOR operation. Participants explore the implications of this operation on the nature of the resulting set, particularly in terms of convexity and representation as linear inequalities.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents two sets of inequalities (A and B) and seeks assistance in combining them into a new set C using the XOR operation.
  • Another participant questions the meaning of the symbol $\oplus$ used by the original poster.
  • The original poster clarifies that $\oplus$ refers to the XOR operation.
  • A later reply discusses the properties of the solution sets of inequalities, noting that they are convex and that combining two convex sets through symmetric difference can lead to a non-convex set, which may not be representable as a set of linear inequalities.

Areas of Agreement / Disagreement

Participants express differing views on the implications of using the XOR operation to combine the sets, particularly regarding the convexity of the resulting set and its representation as linear inequalities. The discussion remains unresolved.

Contextual Notes

The discussion does not clarify the specific conditions under which the symmetric difference may or may not yield a representable set of linear inequalities, leaving some assumptions and definitions implicit.

Barioth
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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!
 
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re: Two inequation set into one.

What do you mean by $\oplus$?
 
re: Two inequation set into one.

the XOR operation, sorry I should have said so!
 
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.
 

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