Can any set be a proper subset of itself & please give an example of why or why not?As we know, a set is just a collection of objects that are similar in some way, like a gaggle of geese, pride...

Can any set be a proper subset of itself & please give an example of why or why not?

As we know, a set is just a collection of objects that are similar in some way, like a gaggle of geese, pride of lions, or an army of ants.

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mfonda's profile pic

Matthew Fonda | eNotes Employee

Posted on

First, let's consider the definition of a subset and a proper subset: We say that a set A is a subset of a set B if every element in A also exists in B. We say that A is a proper subset of B if A is a subset of B and there exists at least one element in B that does not exist in A.

A set cannot be a proper subset of itself.

Proof:

Let A be a set. Suppose, for contradiction, that A is a proper subset of itself. By definition of proper subset, then there exists some element in A that does not exist in A. Therefore a set cannot be a proper subset of itself.

 

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jueli | High School Teacher | (Level 1) eNoter

Posted on

No set is a proper subset of itself because, by definition, a "proper subset" has fewer elements than the original set.

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