Trying to understand specific condition of a set

As requested:

What I am asking, can {x} ∈ S and {x} ⊊ S, where x is a integer value and S is the set?

The problem is:

Find a pair set S such that {x} is an element of S and {x} is not included in S.

My answer is, that this is not possible.

From what I know:

It is possible to have a set, A, and second set, B. Set B can be contained as an element of set A, however that also means that any element contained in set B is included in A. So A={B} while B={0, 3}, then A = {{0,3}}.

Is my logic correct?

--- Original

Sorry, but I am a mature student taking this course Discrete Structures for the first time and am having a hard time trying to understand a particular situation dealing with sets.

What I am feeling that I should know, to answer a particular question, is if a element can be apart of a set but not included within it. I clearly do not understand my notes, practice and reading.

Can someone explain to me if this situation is possible or not?

Re: Trying to understand specific condition of a set

Quote:

Originally Posted by

**navitude89** What I am feeling that I should know, to answer a particular question, is if a element can be apart of a set but not included within it. I clearly do not understand my notes, practice and reading.

I have absolutely no idea what *"if a element can be apart of a set but not included within it."* could possibly mean.

Can you clear up its meaning? Maybe give some examples.

Re: Trying to understand specific condition of a set

Quote:

Originally Posted by

**navitude89** As requested:

What I am asking, can {x} ∈ S and {x} ⊊ S, where x is a integer value and S is the set?

The problem is:

Find a pair set S such that {x} is an element of S and {x} is not included in S.

My answer is, that this is not possible.

Well that is better. But still this is confused: *{x} is an element of S and {x} is not included in S*.

Does that mean $\displaystyle \{x\}\in S$ BUT $\displaystyle \{x\}\not\subset S~?$

Let $\displaystyle \mathbb{Z}$ be the set of integers and $\displaystyle \mathcal{P}(\mathbb{Z})$ be the powerset of $\displaystyle \mathbb{Z}$( set of all subsets).

If $\displaystyle S=\mathbb{Z}\cup\mathcal{P}(\mathbb{Z})$ then it has the property that $\displaystyle (\forall n\in\mathbb{Z})[n\in S~\&~\{x\}\in S]$.

Please **do not** edit a post to clarify. Post a new reply please.

Re: Trying to understand specific condition of a set

" \{x\}\in S BUT \{x\}\not\subset S~" that is what I am trying to understand, however I do not understand anything that you have provided underneath.

I am sorry for taking your time. Can you please close this thread.

Re: Trying to understand specific condition of a set

Quote:

Originally Posted by

**navitude89** " \{x\}\in S BUT \{x\}\not\subset S~" that is what I am trying to understand, however I do not understand anything that you have provided underneath. I am sorry for taking your time. Can you please close this thread.

Let $\displaystyle S=\{\1,\{2\}\}$ now $\displaystyle \{2\}\in S$ but $\displaystyle \{2\}\not\subset S$ because $\displaystyle 2\notin S$.