Proving Boundary of Union is Subset of Union of Boundaries

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In summary, the conversation is discussing how to prove that the boundary of the union of two sets is a subset of the union of the boundaries of the two sets. One approach suggested is to use the identity A c= B <==> cl(B) c= cl(A) and set algebra, but the speaker wonders if there is an easier way. Another approach is proposed, which involves using the concept of closure and neighborhood to prove that x must be a boundary point of X or Y. The speaker also clarifies a confusion about the complement symbol and ultimately solves the problem using the set algebra approach.
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Homework Statement



I'm trying to prove that the boundary of the union of two sets is a subset of the union of the boundaries of the two sets. i.e. Show that B(X u Y) c= B(X) u B(Y).

Homework Equations





The Attempt at a Solution



Since I didn't know what to do here, I used the identity A c= B <==> cl(B) c= cl(A), and then a lot of set algebra to prove it. Given how simple the statement looks, I'm wondering if there is an easier way. Thanks.
 
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  • #2
I don't see any need to deal with the closure. If x is a member of B(X \cup Y), what does that tell you about x? Use that to prove it must be a boundary point of X or a boundary point of Y.
 
  • #3
Actually, didn't know what closure is, so confused the cl symbol I've seen, with complement. So, yea, when I wrote cl I meant complement.

O let me try this. If x in B(X u Y) then for all r>0 the r-neighbourhood of x contains at least some point p in X u Y and another point q in C(X u Y) , (C stands for complement). But now I'm having difficulty with the case where p in X but q in C(Y). This gives the possibility that x is in none of the boundaries of the sets. But if you say that x must be in the boundaries of the sets, then there is no need to solve the problem.
 
  • #4
Im good, got it, just had to use C(X u Y) = C(X) n C(Y), so now it is clear what hapens to q. thank you.
 

1. What is the definition of "Proving Boundary of Union is Subset of Union of Boundaries"?

This refers to the mathematical concept of showing that the boundary of the union of two sets is a subset of the union of the boundaries of those two sets.

2. Why is this concept important in mathematics?

This concept is important because it helps us understand the relationship between the boundaries of sets and their unions. It allows us to make conclusions about the boundaries of combined sets based on the boundaries of their individual parts.

3. How do you prove this statement?

To prove that the boundary of the union is a subset of the union of boundaries, you would need to show that any point in the boundary of the union must also be in the union of boundaries. This can be done using logical arguments and mathematical proofs.

4. Are there any real-life applications of this concept?

Yes, this concept is used in various fields such as computer science, engineering, and physics. For example, in computer graphics, this concept is used to determine the boundary of a composite shape made up of multiple smaller shapes.

5. Is this concept limited to two sets only?

No, this concept can be extended to any number of sets. The statement would then be "Proving Boundary of Union is Subset of Union of Boundaries of Multiple Sets". The same principles and methods would apply for proving this statement.

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