Quote:
Originally Posted by furyshade
that's what i thought, thanks. also i have a problem involving showing if a set is transitive, "a set A is transitive if each element of A is also a subset of A". maybe i am missing something but i don't see how a set could not be transitive. could someone help me out and maybe give an example of a non-transitive finite set?
Let 0 = {}; then, 0 is "vacously" a transitive set. { {} } = {0} is also a transitive set since 0 = {} is a subset of
any set. One can define 1 as {0} and define 2 as {0,1} = { {}, { {} } }, etc.
On the other hand, {1} = { {0} } = { { {} } } is not transitive since 1 = { {} }, the only element, is not a subset of {1}; the only subsets of {1} are {1} and 0 = {}.
There are a lot of "mundane" examples too. For example, any nonempty set S of objects that are not sets will do since each object (by virtue of not being a set) can not be a subset of S.