Understanding Subsets and Elements: Examining the Truth of ∅∈{0}

[Bashyboy]

Homework Statement

The question asks me to determine whether the statement is true or false, the statement being .∅∈{0}

Homework Equations

The Attempt at a Solution

I said that the statement was true, but apparently it is false. Wouldn't a set such as {1,{1}} be made up of the elements 1 and {1}, having a set as one of its elements? So the set in the problem could be written as, {0, ∅}, meaning that the null set is a subset, and also an element?

 

[LCKurtz]

Homework Statement

The question asks me to determine whether the statement is true or false, the statement being .∅∈{0}

Homework Equations

The Attempt at a Solution

I said that the statement was true, but apparently it is false. Wouldn't a set such as {1,{1}} be made up of the elements 1 and {1}, having a set as one of its elements? So the set in the problem could be written as, {0, ∅}, meaning that the null set is a subset, and also an element?

{0} is a set containing a single element. You have $0 \in \{0\}$ but that is the only element. The null set is not a member of this set. The only subsets of {0} are itself and the empty set ∅. It would be correct to write $\Phi \subset \{0\}$.

 

[Bashyboy]

How about {1, {1}}, what are the elements of this set? 1 and {1}?

 

[STEMucator]

How about {1, {1}}, what are the elements of this set? 1 and {1}?

This is true ^. Also, the empty set is a proper subset in regards to your question.

 

[Bashyboy]

Okay, so then sets can also be elements, except for the null set?

 

[Dansuer]

The null set can be an element of a set. Example: {1,{1},∅}. Now the null set is both an element and a subset of that set.

 

[Bashyboy]

So, then the book's answer to the question in the original post was wrong? Or does the null set have to be explicitly written, like you, Dansuer, did?

 

[Dansuer]

Yes, it needs to be explicitly written. {0} ≠ {0, ∅}. ∅ is an element of the second set but not of the first. While ∅ is a subset of both sets.
If something is a subset of a set, it does not mean it is an element of that set.

For example this is wrong:
{1,2,3} is a subset of {1,2,3,4} and so we write {1,2,3,4,{1,2,3}} and we say that {1,2,3} is an element of the set. This is obliviously wrong, but when we are dealing with the null set it's more confusing.
We usually represent a set with {} while we represent the empty set with just ∅. So ∅ might appear to be an element, while actually it a set.

 

[Bashyboy]

Oh, okay, I understand. Thank you.