Are These Set Closure and Interior Properties True or False?

[proplaya201]

Homework Statement

6) Prove or give a counter-example of the following statements
(i) (interiorA)(closure) intersect interior(A(closure)):
(ii) interior(A(closure)) intersect (interiorA(closure)):
(iii) interior(A union B) = interiorA union interiorB:
(iv) interior(A intersect B) = interiorA intersect interiorB:
(v) (A union B) (closure) = (closure) A union (closure)B:
(vi) (A intersection B)(closure) = (closure) A intersection (closure)B:

Homework Equations

The Attempt at a Solution

i just don't know how to do any of them

 

[Dick]

Your first two aren't even statements. Aren't you supposed have an = sign or something? So we'll pass on those. And it's pretty weak to say "no idea on ANY of them". Try iii). Suppose A=[0,1] and B=[1,2]. Conclusion? Try to find a counterexample first. If you can't find a counterexample, then try to prove it. Some of those are not all that hard.

 

[proplaya201]

my mistake
for the first one i meant to say write

the closure (interior A) is a proper subset of the interior (closure A)

i took A = [1,2]

then the right side would equal [1,2] and the left would be (1,2).

therefore

the (closure (interior A)) is not a proper subset of the (interior (closure A)), right?

 

[Dick]

my mistake
for the first one i meant to say write

the closure (interior A) is a proper subset of the interior (closure A)

i took A = [1,2]

then the right side would equal [1,2] and the left would be (1,2).

therefore

the (closure (interior A)) is not a proper subset of the (interior (closure A)), right?

That's the idea. Just remember if it had come out to be true, that doesn't prove it's true in all cases.