What is an empty family of subsets?

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SUMMARY

The discussion centers on the concept of an empty family of subsets within the context of set theory, specifically regarding the set of real numbers ℝ. Participants clarify that an empty family, by definition, contains no elements, and therefore cannot contain any subsets of ℝ. The confusion arises from the terminology used in the book, which states "Let A be an empty family of subsets," leading to questions about the necessity of specifying subsets when the family is empty. The consensus is that the phrase is technically accurate but may be misleading.

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Aziza
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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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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.
 
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 ??
 

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