Question

State the most general conditions on the subset A of U if A' intersect U = empty set.

Answer 1

If A' intersect U = empty set, then A = U

Answer 2

If A' intersect U = empty set, then A = U


Proof:


Proof by contradiction


Let's say set A isn't the universal set. This will be our assumption that we'll contradict.


So set A would be some smaller subset of set U.


This would mean that there exists elements that are in set U, but NOT in set A. These elements would be part of set A' instead.


So because these elements are in BOTH set A' and set U, this means that the intersection of these two sets is NOT the empty set.


But this is a contradiction. We're told that A' intersect U = empty set, so this is where we flip our assumption from "A isn't the universal set" to "A is the universal set"



Visually, imagine this rectangle is the universal set U



Inside the rectangle is the set A (represented by the circle) which right now we're going to make smaller than U


Anything inside the rectangle but not inside the circle will be the set A'


Since A is smaller than U, we can see that elements exist in U and A'. But again, A' intersect U = empty set means that sets A' and U have nothing in common. So this picture is clearly wrong.


The only way to fix it is to make set A the same size as set U. Basically, make set A equal to set U.