Set theory terms

[SenatusPopulusqueRomanus]

Hi everyone,
I'm just wanting to read up on how set theory defines properties and the relationships between sets and members in terms of properties.

Specifically, could anyone tell me the name of the term that describes how members of a set that has properties (a, b, c) might "inherit" some of those properties, but not others? Or even what the correct term for "inherit" would be in this context (distribution?).

Another thing I'm wanting to read about is the laws of containment in set theory, so one set being contained by, or subsidiary to, another.

I'm basically just looking for terms and resources so I can read up, but I cant find where to start.

Thanks for any help

 

[blamocur]

This request seems rather vague. I'd have searched the internet, then come up with more specific questions. My quick search for "set theory properties inheritance" came back with a list which included, among others, https://en.wikipedia.org/wiki/Hereditary_property and http://jdh.hamkins.org/local-properties-in-set-theory/ -- do you see anything useful in there?

 

[pka]

I'm just wanting to read up on how set theory defines properties and the relationships between sets and members in terms of properties.
Specifically, could anyone tell me the name of the term that describes how members of a set that has properties (a, b, c) might "inherit" some of those properties, but not others? Or even what the correct term for "inherit" would be in this context (distribution?).
Another thing I'm wanting to read about is the laws of containment in set theory, so one set being contained by, or subsidiary to, another.
I'm basically just looking for terms and resources so I can read up, but I cant find where to start.

It is rather easy to find a used edition in good condition of Paul Halmos' Naïve Set

 

[SenatusPopulusqueRomanus]

This request seems rather vague. I'd have searched the internet, then come up with more specific questions. My quick search for "set theory properties inheritance" came back with a list which included, among others, https://en.wikipedia.org/wiki/Hereditary_property and http://jdh.hamkins.org/local-properties-in-set-theory/ -- do you see anything useful in there?

Those look like quite good starting points - "transitivity" seems like it might be or lead to what I'm looking for. I actually found the wikipedia page for subset which already has some of what I was looking for. Thank you