# Subsets and Elements Of

1. Sep 27, 2012

### Bashyboy

1. The problem statement, all variables and given/known data
The question asks me to determine whether the statement is true or false, the statement being .∅∈{0}

2. Relevant equations

3. 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?

2. Sep 27, 2012

### LCKurtz

{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 $\Phi \subset \{0\}$.

3. Sep 27, 2012

### Bashyboy

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

4. Sep 27, 2012

### Zondrina

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

5. Sep 28, 2012

### Bashyboy

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

6. Sep 28, 2012

### 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.

7. Sep 28, 2012

### 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?

8. Sep 28, 2012

### 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.

9. Sep 28, 2012

### Bashyboy

Oh, okay, I understand. Thank you.

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