Help~find the interior, boundary, closure and accumulation points of the following.

In summary, the conversation discusses different mathematical concepts such as natural numbers, irrational numbers, and straight lines in R^n. The sets for part a and b are determined, with set A being bounded and having accumulation points near 2 and any number of the form 1+1/m between 1 and 2. For part b, the set is interpreted as all irrational numbers less than the square root of 2 and all natural numbers. In part c, the straight line L through 2 points a and b in R^n is illustrated using the definition of a neighborhood.
  • #1
ShengyaoLiang
23
0
a. 1/n + 1/m : m and n are both in N
b. x in irrational #s : x ≤ root 2 ∪ N
c. the straight line L through 2points a and b in R^n.


for part c. i got: intA= empty ; bdA=clA=accA=L Is this correct? how about part a and part b...i am so confused...
 
Last edited:
Physics news on Phys.org
  • #2
As far as the first is concerned I don't know because you didn't say what L is! For (b) you need to know that "between any two irrational numbers, there is at least one rational number". (c) should be easy using the definition of "neighborhood". What does a neighborhood in Rn look like?
 
  • #3
your c) looks correct.

draw pictures. it will help get rid of the confusion. what textbook are you using for this class?

now for a) and b) determine the sets. Is set A) bounded?

Hint for a's accumulation points, how many points come "near" 2? how about ANY number of the form 1+1/m in between 1 and 2? Fix n=1, let m=1,2,3..., what happens? Fix n as N (N is any fixed integer) and let 1/N +1/m with m=1,2,3... what happens? All these sequences I have suggested are contained in the set A.

for b) do you mean all irrational numbers that are less than the root of 2 and all irrationals that are natural numbers?

edit: werever i say integer, i mean positive integer!
 
  • #4
don't have a formal texeboot for analysis1, only have a courseware...

thanks a lot.
 
  • #5
SiddharthM said:
your c) looks correct.
for b) do you mean all irrational numbers that are less than the root of 2 and all irrationals that are natural numbers?

That would make the second part pretty meaningless!

I would interpret it as b. {x in irrational #s : x ≤ root 2} ∪ N
 
  • #6
sorry yes, that is actually what i meant to ask.
 

1. What is the difference between interior, boundary, closure, and accumulation points?

The interior points of a set are the points that are entirely contained within the set. The boundary points are the points that lie on the edge or boundary of the set. The closure of a set includes all of the interior points and the boundary points. The accumulation points are the points that are infinitely close to the set, but may not be part of the set itself.

2. How do you find the interior points of a set?

To find the interior points of a set, you need to determine which points are completely surrounded by the set. This can be done by drawing a shape or graph of the set and identifying which points are inside the set and not on the boundary.

3. How are boundary points different from interior points?

Boundary points are points that lie on the edge or boundary of a set, while interior points are points that are completely contained within the set. Boundary points can be thought of as the "border" between the inside and outside of a set.

4. How can you determine the closure of a set?

The closure of a set can be determined by including all of the interior points and the boundary points of the set. This can be done by examining the set and identifying which points are not included in the interior, but are still part of the set.

5. What is the significance of accumulation points in a set?

Accumulation points are important because they represent points that are infinitely close to the set, but may not be part of the set itself. These points can help us understand the behavior and limits of the set, and are often used in calculus and other mathematical concepts.

Similar threads

  • Topology and Analysis
Replies
2
Views
1K
  • Topology and Analysis
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
Replies
2
Views
1K
  • Special and General Relativity
Replies
7
Views
998
Replies
2
Views
1K
Replies
86
Views
5K
  • Calculus and Beyond Homework Help
Replies
4
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
5K
  • Introductory Physics Homework Help
Replies
11
Views
2K
Back
Top