Trying to understand specific condition of a set

**navitude89** · January 22nd 2013, 01:16 PM

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?

**Plato** · January 22nd 2013, 01:36 PM

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.

**Plato** · January 22nd 2013, 02:27 PM

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 $\{x\}\in S$ BUT $\{x\}\not\subset S~?$

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

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

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

**navitude89** · January 22nd 2013, 02:36 PM

" \{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.

**Plato** · January 22nd 2013, 02:46 PM

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.