Let A,B, and C be sets.?

Prove that A is a subset of B iff A-B= empty set.

Let A,B, and C be sets.?
Let "n" denote intersection, "u" denote union, and B' be the complement of B. By definition, A-B = A n B'





1. Proving =%26gt;


Start with letting A be a subset of B.


Let's try using proof by contradiction here.


Suppose that A-B is nonempty.


Then there exisits an element x such that x is in A-B.


That means that x is in A n B'.


That means that x is in A and x is in B'


Since x is in A, and A is a subset of B, then x is in B.


This is a contradiction, because x cannot be in both B and B'.


So that must mean that A-B = empty set.





2. Proving %26lt;=


Now suppose that A-B = A n B' = empty set.


Let x be in A.


Then x is in (A n B) u (A n B') since A = (A n B) u (A n B').


Then either x is in (A n B) or x is in (A n B').


However x cannot be in A n B' since it is the empty set.


So that must mean that x is in A n B.


Which means that x is in B.


That means that A is a subset of B.
Reply:A-B is well known as A n B'. You should know that, Dr. D. Report It

Reply:A is a subset of B if


A intersect B = A





AunionB = A + B - AintersectB


If A is a subset of B, then


AunionB = B





I'm not sure what your term A-B means.