/sci/ - Science & Math

>>14977059
This is an interesting question
>>14977221
S/E is the set of all equivalence classes of the elements of S under the equivalence relation E
S/E in fact forms a partition on S, so that [eqn] \bigcup S/E = \bigcup_{ x \in S/E} x = S [/eqn]
One can also easily prove that S has cardinality [math] 2^{\aleph_0} [/math] (see >>14977135)
Interestingly the cardinality of every equivalence class is [math] \aleph_0 [/math], if we assume there are only countably many such equivalence classes then we get a contradiction because a countable union of countable sets is also countable but the union of S/E is S which is uncountable, Hence S/E cannot be countable so [math] \aleph_0 < \operatorname{card}(S/E) [/math] however i'm not sure where to go from there.