What is an empty family of subsets?

In summary, the conversation discusses the concept of an empty family and its relation to sets and subsets. The speaker questions how an empty family can contain subsets of a set, and the responder clarifies that an empty family cannot contain any elements, including subsets. The speaker then asks why the book specifies subsets in the definition of an empty family, to which the responder does not have an answer.
  • #1
Aziza
190
1
At the beginning of a question my book is saying: "Let the universe of discourse be the set ℝ of real numbers, and let [itex]A[/itex] be the empty family of subsets of ℝ.."

How on Earth can an empty family contain sets which are subsets of something? A family is a set whose elements are sets. If the family is empty, it cannot contain any elements. Thus it cannot contain sets which are subsets of ℝ! Am I right?
 
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  • #2
Aziza said:
How on Earth can an empty family contain sets which are subsets of something?
It doesn't. That's why it's an empty family.
 
  • #3
Hurkyl said:
It doesn't. That's why it's an empty family.

Then why can't my book just say, Let A be an empty family. Why is it saying, Let A be an empty family of subsets ??
 

1. What is an empty family of subsets?

An empty family of subsets refers to a collection of subsets that contains no elements. In other words, it is a set that has no members or elements.

2. How is an empty family of subsets different from an empty set?

An empty family of subsets is a collection of subsets, while an empty set is itself a subset. The main difference is that an empty family of subsets can still contain subsets with no elements, while an empty set cannot contain any elements at all.

3. What is the purpose of an empty family of subsets?

An empty family of subsets may seem trivial, but it serves an important role in mathematics and set theory. It allows for the existence of certain mathematical concepts, such as the universal set and the power set, which are essential for understanding and proving many mathematical theorems.

4. Can an empty family of subsets be considered a set?

Yes, an empty family of subsets can be considered a set, as it follows the definition of a set as a collection of objects. However, it is important to note that it is a special type of set with no elements.

5. How is an empty family of subsets represented in mathematics?

In mathematics, an empty family of subsets is often denoted by the symbol ∅ or {} (empty curly braces). It can also be represented using set-builder notation as {x | x is a subset of a set and x has no elements}.

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