/sci/ - Science & Math

[Anonymous No.12085173]

Is it possible to have an arbitrary member x of set A such that x is not a subset of set A?

Or is it that for any arbitrary member x of A, x is always a subset of A?

I know that a set containing arbitrary member x of A is not a member of set A, unless A contains a set containing arbitrary member x of A.

[Anonymous No.12085302]

[math]x \in A[/math] means that [math]\{x\} \subseteq A[/math] but doesnt necessarily mean that x itself is a subset. It technically could be if you have a set like [math]\{ x,\{ x \} \} [/math]

[Anonymous No.12085312]

The element x of A is not a subset of A because it is not a set.

The set {x} is a subset A. Although in elementary (naive) set theory this situation technically doesn't arise because sets can only contain elements that are themself sets.

Insofar as you can have 'arbitrary elements' a.la the envelope paradox.

[Anonymous No.12085501]

>>12085302
>>12085312
So then, if A is a set containing the set X = {{}}, and assuming that {} is a subset of A and X,
{} is a member and subset of X, and a subset but not a member of A.

X is a member of A, and {X} is a subset of A, X is a subset of itself, but X is not a subset of either {X} or A?

If I did not misunderstand,

Then how in the fuck does the ∀x(x∈Ax⊆A) definition of transitivity work? (This is the definition provided to me in my introductory text)
How could it exist?
I'm so fucking stupid, anons.

[Anonymous No.12085521]

>>12085501
I guess the arrow for the boolean "conditional" operator doesnt render.
"For all x if (If x∈A then x⊆A is true)" is what I was provided as the "definition of transitivity".