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:=
\(\displaystyle X_1-X_2 \leq 1\)
\(\displaystyle X_1 \leq3\)
\(\displaystyle X_2 \geq 1\)
\(\displaystyle X_1,X_2 \geq 0\)

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]

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.

 

[blue_raver22]

Hello! It seems like you are trying to combine two sets of inequalities into one set. This can be accomplished by using set operations, such as the union or intersection of sets. In this case, you are using the exclusive or operation (symbolized by the symbol ⊕) to create a new set C.

To tackle this problem, you can start by looking at each individual inequality and determining how it relates to the other inequalities. For example, in set A, we have X_1-X_2 ≤ 1 and X_1 ≤ 3. This means that X_1 must be less than or equal to 3 and X_2 must be greater than or equal to 2 (since X_1 - X_2 ≤ 1 and X_1 ≤ 3, we can substitute X_1 = 3 and solve for X_2 to get X_2 ≥ 2). Similarly, in set B, we have X_1 + X_2 ≥ 5 and X_1 ≤ 5. This means that X_1 must be less than or equal to 5 and X_2 must be greater than or equal to 0 (since X_1 + X_2 ≥ 5 and X_1 ≤ 5, we can substitute X_1 = 5 and solve for X_2 to get X_2 ≥ 0).

Now, to combine these two sets using the exclusive or operation, we can create a new set C with the following inequalities:

X_1 ≤ 3
X_2 ≥ 2
X_1 ≥ 4
X_2 ≤ 4

These inequalities satisfy both sets A and B, since they include all the possible values for X_1 and X_2 that satisfy the original inequalities. Therefore, C:= A⊕B.

I hope this helps! Let me know if you have any further questions.