Something to be a subset of something?

[Kamataat]

The book I'm reading says that from $$p \in A$$ and $$A \in M$$ it does not follow that $$p \in M$$, if $$M$$ is a family of sets and $$p$$ is an element of $$A$$.

However, then further down on the same page it says that for any sets $$A, B, C$$ it is true that if $$A \subseteq B$$ and $$B \subseteq C$$, then $$A \subseteq C$$.

What's the difference between the two? Let's say I consider $$A$$ to be an element of $$B$$, then according to the first example, it does not follow that $$A \in C$$.

What's the difference between considering something to be an element of something else and something to be a subset of something?

- Kamataat

 

[phoenixthoth]

The notion of "is a subset of" is based on the notion of "is an element of."

A is a subset of B means that for all x, if x is an element of A then x is an element of B.

Armed with this definition, you can prove the statements you made above.

Let C={{{1}}}, B={{1}}, and A={1}. Then A is an element of B and B is an element of C yet A is not an element of C. If A were an element of C, C would look like this:
{{{1}},{1}}={B,A}.

Next, you can prove that if A is a subset of B and B is a subset of C then A is a subset of C. Let x be an arbitrary element of A. Then x is in B since A is a subset of B. Then since x is in B, as B is a subset of C, x is in C.

 

[jcsd]

Consider the sets A = {p,q}, B = {p,q,r}, C = {p,q,r,s,t}, D = {A,B,C}

Now p is a member of A and A is a member of D, yet p is not a member of D as the only three elements of D are A, B and C.

A is a subset of B as B contains all the elements in A and B is a subset of C as C contains all the elemnts in B, cleraly C must contain all the elements in A and thus A is also a subset of C.

 

[AKG]

No, A is not an element of B, it is a subset. The two are entirely different things. You could say $p \in A$ or $\{p\} \subseteq A$, but not $p \subseteq A$. It won't make sense. Sets and the elements they contain are different types of things, and so the relations involving subsets are different from relations involving elements. B, which is a set (or family) of sets contains sets as elements, and, obviously, sets as elements, but they are still different, since it's subsets are still sets of sets, and its elements are "regular" sets containing "regular" elements like p.

 

[Kamataat]

So it has to do in a way with the fact that $$p \neq \{p\}$$?

- Kamataat

 

[AKG]

Correct...

 

[Kamataat]

Thank you very much for making this clear to me, everyone!

- Kamataat