Rectangle question and closure of the interior?

In summary, the conversation is discussing how to prove that the closure of the interior of a rectangle Q is equal to the closure of Q itself. The speaker suggests using the definition of closure, Cl(A) = int(A) U bd(A), and showing that Cl(int(Q)) = int(int(Q)) U bd(int(Q)). They also mention the difficulty in proving this, and suggest showing that Int(AxB) = Int(A)xInt(B) and that bd(int(Q)) is equal to the set of boundary points {a1, b1, ..., an, bn}.
  • #1
Buri
273
0
The question says:

Show that if Q = [a1,b1]x...x[an,bn] is a rectangle, the Q equals the closure of Int Q.

The definition of closure that I have is Cl(A) = int(A) U bd(A). So I'd like to show that Cl(int(Q)) = int(int(Q)) U bd(int(Q)).

But this just seems to be obvious to me which just makes it hard to prove - I just don't know what to write. Any hints/ideas on how to prove this rigorously?

EDIT!

I guess I'd have to show something like:

Int(AxB) = Int(A)xInt(B)

And then I guess, I'd make a claim that bd(int(Q)) = {a1,b1,...,an,bn} and prove this by showing that no other boundary points exist?

Questions like this I always find hard.
 
Last edited:
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  • #2
You are being asked to show that Q = Cl (IntQ)
You know how to define Int and Bd, so write out the definitions explicitly (in set theory notation) to show that they coincide.
 

Related to Rectangle question and closure of the interior?

1. What is a rectangle question and closure of the interior?

A rectangle question and closure of the interior refers to the process of finding the smallest rectangle that completely encloses a given shape or set of points, and then determining the interior points within that rectangle. This is often used in geometric and mathematical problems to find the boundaries and area of a shape.

2. How is a rectangle question and closure of the interior solved?

To solve a rectangle question and closure of the interior, you first need to plot or visualize the given shape or points on a graph. Then, draw a rectangle that encloses the shape or points. Next, determine the coordinates of the corners of the rectangle and use them to find the length and width. Finally, identify and list all the interior points within the rectangle.

3. What is the importance of solving a rectangle question and closure of the interior?

Solving a rectangle question and closure of the interior is important in mathematics and geometry because it helps us understand and analyze the properties of shapes and points. It also allows us to find the area and perimeter of a given shape, which is useful in real-life applications such as construction, design, and engineering.

4. Can a rectangle question and closure of the interior be solved for any shape?

Yes, a rectangle question and closure of the interior can be solved for any shape, as long as the shape can be enclosed in a rectangle. This includes regular and irregular polygons, circles, and even more complex shapes. The process may vary slightly depending on the shape, but the basic steps remain the same.

5. Are there any real-world examples of using rectangle question and closure of the interior?

Yes, there are many real-world applications of rectangle question and closure of the interior. For example, architects and engineers use this concept to determine the area and boundaries of a building site. Surveyors use it to map out land and determine property lines. It can also be used in computer graphics to determine the boundaries of an image or shape.

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