Show that A=B when A,B are contained in M with d(A,B) <= 3/4

Question

Let U be a non empty set and let M=the set of all subsets of U. Define the metric d on M by d(A,B)=|A("delta")B|.

I'm having trouble knowing where to start with this. I know all of the requirements for it to be a metric but this is different compared to anything my teacher has showed me

Answer 1

Our distance function only takes on non-negative integer values. So if $d(A,B)\le \frac{3}{4}$, then $d(A,B)=0$. And it is easy to verify that the distance between two sets is $0$ precisely if the sets are equal. For $A\Delta B=\emptyset$ if and only if $A=B$.