How Can Two Inequality Sets Be Combined into One?

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SUMMARY

The discussion focuses on combining two sets of inequalities, A and B, into a new set C using the XOR operation (denoted as A ⊕ B). Set A consists of four inequalities involving variables X1 and X2, while set B includes five inequalities. The key conclusion is that the resulting set C may not always be representable as a set of linear inequalities due to the non-convex nature of the symmetric difference operation applied to convex sets.

PREREQUISITES
  • Understanding of linear inequalities and their graphical representation
  • Familiarity with convex and non-convex sets in mathematics
  • Knowledge of set operations, particularly symmetric difference
  • Basic algebra involving multiple variables
NEXT STEPS
  • Study the properties of convex sets and their intersections
  • Learn about symmetric difference in set theory and its implications
  • Explore methods for visualizing solutions to systems of inequalities
  • Investigate applications of linear programming in optimization problems
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Mathematicians, students studying linear algebra, and anyone interested in optimization and set theory will benefit from this discussion.

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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