Constructing a Set with the following characteristics

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In summary, the task is to construct a set ScR such that its interior, closure, closure of interior, and interior of closure are all distinct. This seems impossible as a set, its interior, and its closure cannot all be distinct. The closest solution is the set of rational numbers, but even that does not satisfy all the conditions.
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Homework Statement



Construct a set ScR such that S, the interior of S, the closure of S, the closure of the interior of S, and the interior of the closure of S are all distinct (ie no 2 of them are equal)


Homework Equations


closure of S - smallest closed set containing S
interior of S - set of all points in S for which S is a neighbourhood



The Attempt at a Solution



I am really having trouble starting this...or more precisely, I'm having trouble seeing how this question is possible at all...How can a set, its interior, and its closure all be distinct? If a set is open, then its interior is simply equal to the set itself, so that leaves me with only closed sets to consider. But, if a set is closed, then it contains all of its boundary points, so the closure of S is equal to S...
so the only exception i could think of is the open/closed sets...

but so far the closest I've gotten to something that satisfies all of those things is the set of rational numbers
S = Q
int S = empty set
Closure of S = Real number line
but then closure of the interior of S = closure of the empty set = empty set
so that's equal to int S and doesn't work
and the interior of the closure of S = real number line = closure of S so that doesn't work either...help!
 
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I just finished answering this question from someone else. You can find the thread here:
<https://www.physicsforums.com/showthread.php?t=314690>
Enjoy!
 
  • #3
Oh wow...awesome! thanks :) (its probably a classmate of mine haha...since both of us thought of rational numbers, and that's something we discussed in class)
 

1. How do you construct a set with specific characteristics?

To construct a set with specific characteristics, you first need to determine what those characteristics are. This may include the number of elements in the set, the types of elements, or any other defining features. Once you have identified the characteristics, you can use mathematical operations or logical statements to create the set. For example, if you want to construct a set of even numbers between 1 and 10, you can use the notation {x | x is an even number between 1 and 10}.

2. What is the difference between a finite and infinite set?

A finite set is a set that has a limited or countable number of elements, while an infinite set has an uncountable number of elements. For example, the set {1, 2, 3, 4, 5} is finite because it has 5 elements, while the set of all even numbers is infinite because it has an infinite number of elements.

3. Can a set have duplicate elements?

No, a set cannot have duplicate elements. A set is defined as a collection of distinct objects, so each element must be unique. If a set were to contain duplicate elements, it would no longer be considered a set. However, it is possible for two different sets to have the same elements.

4. How do you determine if an element belongs to a set?

To determine if an element belongs to a set, you can use the notation x ∈ S, where x is the element and S is the set. This notation means "x is an element of the set S." If the element is included in the set, the statement is true, and if the element is not included, the statement is false.

5. What is the cardinality of a set?

The cardinality of a set is the number of elements it contains. For example, the set {1, 2, 3, 4, 5} has a cardinality of 5, while the set of all even numbers has an infinite cardinality. The cardinality of a set can be denoted by |S|, where S is the set.

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