The intersection of an empty collection of subsets of X is equal to X?

In summary, the conversation revolves around the intersection of an empty collection of subsets of a set X, which is equal to X. The speaker finds this concept counter-intuitive and asks for insight, to which another person responds by explaining that in the ultimate case of taking the fewest sets, the intersection is as big as possible.
  • #1
ModernLogic
Hi,

I'm reading HL Royden's real analysis, though my question pertains more to set theory.

Let X be a set. Then the intersection of an empty collection of subsets of X is equal to X. I understand this is not an intersection of empty subsets but it is still very counter-intuitive. Can anyone provide insight?
 
Physics news on Phys.org
  • #2
Well, what element wouldn't be in the intersection?
 
  • #3
Consider an intersection of lots of sets. If you take fewer sets, the intersection is bigger, right? So, in the ultimate case, when you take the fewest sets of all, the intersection is as big as possible.
 

What does it mean for the intersection of an empty collection of subsets of X to be equal to X?

It means that the intersection of all possible subsets of X is equal to the entire set X itself. This is because there are no subsets to intersect, so the intersection is essentially the same as the original set.

Why is the intersection of an empty collection of subsets of X defined as X?

This definition is consistent with the concept of an empty intersection. In mathematics, the intersection of any set with an empty set is always equal to the empty set. In this case, the empty collection of subsets represents an empty set, and the intersection with X is therefore also the empty set.

What is the significance of the intersection of an empty collection of subsets of X?

The intersection of an empty collection of subsets of X is a fundamental concept in set theory. It helps establish the relationship between sets and their subsets, and also demonstrates the concept of an empty set. It is also used in various proofs and mathematical calculations.

Can the intersection of an empty collection of subsets of X be different from X?

No, the intersection of an empty collection of subsets of X will always be equal to X. This is because there are no subsets to intersect, so the result will always be the same as the original set.

How does the intersection of an empty collection of subsets of X relate to the concept of a universal set?

The intersection of an empty collection of subsets of X is equal to X, which is the definition of a universal set. This means that X contains all possible elements, and the intersection of any subsets with X will always result in X. Therefore, the concept of a universal set is closely related to the intersection of an empty collection of subsets of X.

Similar threads

I Help with measurable cardinal
    • Set Theory, Logic, Probability, Statistics
Replies
5
Views
1K
MHB Proving Topology in X: A Look at Union & Intersection
    • Topology and Analysis
Replies
2
Views
1K
What is the definition of the empty intersection?
    • Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
I Direct limit of multiverse models of ZFC
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
MHB Find the intersection value of 3 subsets
    • Set Theory, Logic, Probability, Statistics
Replies
4
Views
4K
Unions and intersections of collections of sets
    • Set Theory, Logic, Probability, Statistics
Replies
4
Views
3K
I Thinking about equality of infinite sets
    • Set Theory, Logic, Probability, Statistics
Replies
5
Views
1K
Disproving A=B with Counter Example: Sets A, B & C
    • Set Theory, Logic, Probability, Statistics
Replies
9
Views
3K
Compactness in Topology and in Logic
    • Set Theory, Logic, Probability, Statistics
Replies
4
Views
3K
Exploring Empty Family of Subsets of \mathbb{R}
    • Set Theory, Logic, Probability, Statistics
Replies
4
Views
3K

Hot Threads

Recent Insights

Back
Top