Is the Union and Intersection of a Null Collection Valid in Set Theory?

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In summary: So (Disjoint) C= S (the original set).In summary, the book states that if C is the null collection of subsets of S, then the union of C is equal to the null set and the intersection of C is equal to the original set S. This is true according to the definition of a null collection of subsets. Additionally, the concept of a null collection of subsets can be compared to a game with no rules, where any move is allowed.
  • #1
JasonRox
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The book that I'm reading is saying...

If C is the null collection of subsets of S then,

(Union) C = Null

(Disjoint) C = S

Is this true?
 
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  • #2
How does your book define a null collection of subsets?
 
  • #3
JasonRox said:
The book that I'm reading is saying...
If C is the null collection of subsets of S then,
(Union) C = Null
(Disjoint) C = S
Is this true?
Take it as a definition, or read this, for example:

http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2004;task=show_msg;msg=0896.0001

since i presume by (disjoint) you actually mean intersection.

incidentally i got that answer by insertingf the words empty intersection into google and clicking the first link.

empty union requires you to follow the third (non indented) link.

you might want to remember that the next time you struggle to check a definition,
 
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  • #4
The truth is that I searched and searched. Then I thought and thought, then searched again.

Using the definition it is true, and I see that, but I was skeptical about it.

Thanks, for the link.
 
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  • #5
imagine a list of rules for a game which has no rules at all!

the intersection of a null collection of sets, corresponds to those moves which satisfy all the rules, hence any move at all, i.e. S.etc...you do the other case
 
  • #7
JasonRox said:
The book that I'm reading is saying...
If C is the null collection of subsets of S then,
(Union) C = Null
(Disjoint) C = S
Is this true?

if x is in (Union)C, then it must be in at least one of the members of C. But C has no members so that is always false. Yes, (Union) C= Null set.

By (Disjoint) C do you mean the intersection[\b] of all the members of C?

Let x be any member of S. If x is NOT in (intersection) C, then there must be some member of C such that x is NOT in it. But that's NEVER true because C has no members! Therefore every member of S is in (Intersection) C.
 

FAQ: Is the Union and Intersection of a Null Collection Valid in Set Theory?

What is Basic Set Theory?

Basic Set Theory is a branch of mathematics that deals with the study of sets, which are collections of objects. It provides a foundation for other areas of mathematics, such as topology and analysis, and is widely used in various fields, including computer science and statistics.

What are the basic concepts in Set Theory?

The basic concepts in Set Theory include sets, elements, subsets, unions, intersections, and complements. A set is a collection of objects, and the objects in a set are called elements. A subset is a set that contains only elements from another set. A union is a set that contains all the elements from two or more sets, and an intersection is a set that contains only the elements that are common to two or more sets. A complement is a set that contains all the elements that are not in a given set.

What is the difference between a finite and infinite set?

A finite set is a set that contains a limited number of elements, while an infinite set contains an infinite number of elements. For example, the set of all natural numbers is an infinite set, while the set of all letters in the English alphabet is a finite set.

What is Topology?

Topology is a branch of mathematics that studies the properties of spaces that are preserved under continuous transformations, such as stretching and bending. It is used to describe the properties of objects that are not affected by continuous deformations, such as a coffee mug and a donut, which are topologically equivalent.

What are the basic concepts in Topology?

The basic concepts in Topology include open and closed sets, continuity, compactness, and connectedness. An open set is a set that contains all its limit points, while a closed set contains all its boundary points. Continuity is a property of functions that preserves the topological structure of the underlying space. A compact set is a set that is closed and bounded, and a connected set is a set that cannot be divided into two disjoint non-empty sets.

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