- #1
1MileCrash
- 1,335
- 41
If I know that A is a subset of B,
what can I say about the relationship between A-complement and B?
what can I say about the relationship between A-complement and B?
If [itex]A\subseteq B\subseteq X[/itex], then [itex]X\setminus A \supseteq X\setminus B[/itex].
Draw a Venn diagram.
There IS no relationship that I can see.
A complement probably contains elements that are in B, and also probably contains elements that are not in B, except for the weird cases where B is the whole universe or A=B. Do you have a specific problem in mind that inspired this question?
I think in this case A is always the empty set. If A is a subset of B then [itex] A\cap B = A [/itex].
From there of course you can prove that [itex] B \subset X\setminus A [/itex] but that's not a very interesting relationship anymore :tongue:
The closure is
[tex] \bigcap_{i\in I} C_i [/tex]
where the C_{i} are the closed sets containing B. Taking complements, let [itex]U_i = X\setminus C_i [/itex] be the open sets which are disjoint from B. Then the closure of B is
[tex] = \bigcap_{i\in I} X\setminus U_i = X\setminus \bigcup_{i\in I} U_i [/tex]
which is exactly how you are calculating the closure - I think your statement is an accurate description of the closure of B.
You might be better served just starting a new thread with your whole proof to figure out where it goes wrong at this point.
The closure is
[tex] \bigcap_{i\in I} C_i [/tex]
where the C_{i} are the closed sets containing B. Taking complements, let [itex]U_i = X\setminus C_i [/itex] be the open sets which are disjoint from B. Then the closure of B is
[tex] = \bigcap_{i\in I} X\setminus U_i = X\setminus \bigcup_{i\in I} U_i [/tex]
which is exactly how you are calculating the closure - I think your statement is an accurate description of the closure of B.
You might be better served just starting a new thread with your whole proof to figure out where it goes wrong at this point.