- #1
geoffrey159
- 535
- 72
Homework Statement
Let ##A,B \in {\cal P}(E)##. Solve in ##{\cal P}(E)## the following equations:
- ##X\cup A = B##
- ##X\cap A = B##
- ##X - A = B##
Homework Equations
The Attempt at a Solution
- We have ##A\cup B = (A\cup X)\cup A = A\cup X = B##. So ##A\subset B## and the solution cannot be less than ## C_B(A) ##. So ## X = C_B(A) \cup G,\ G\in{\cal P}(A) ##
- We have ##A\cap B = A\cap X = B ##. So ##B\subset A## and the solution cannot be less than ##B##. So ## X = B \cup G,\ G\in{\cal P}(C_E(A))##
- We have ##A\cap B = \emptyset## and ## A\cup B = A\cup X##. So the solution can't be less than ##B##. Finally, ##X = B\cup G,\ G\in{\cal P}(A) ##
Is it correct ?