Let A and B be subsets of Rn a) Show that A B) (AC U (B)C) 3A b) Show that if x (A B) and x (A B)U(B A), then x AB c) Prove that A B) (A B) U (B A) U (A B) d) Show that even in R, there exist sets A and B such that (A B) (A n B) (B A) (A B) ...

a Let x A B A c B c Case 1 xA c Since B r x intersects A it foll ...

Let A and B be subsets of Rn. a) Show that {A B) (AC U

Question:

Let A and B be subsets of Rn.
a) Show that ϑ{A ∩ B) ∩ (AC U (ϑB)C) ⊂ 3A.
b) Show that if x ∈ ϑ(A ∩ B) and x ∉ (A ∩ ϑB)U(B ∩ ϑA), then x ∈ ϑA∩ϑB.
c) Prove that ϑ{A ∩ B) ⊂ (A ∩ ϑB) U (B ∩ ϑA) U (ϑA ∩ ϑB).
d) Show that even in R, there exist sets A and B such that ϑ(A ∩ B) ≠ (A n ϑB) ⋃ (B ∩ ϑA) ⋃ (A ∩ ϑB).