# Subsets and Elements Of

Bashyboy

## Homework Statement

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

## 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?

Homework Helper
Gold Member

## Homework Statement

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

## 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}?

Homework Helper
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.