The first one is simple logic.
The statment that "if then " is TRUE.
Recall that a false statement implies any statement and is false.
So both of these are true:
then and then
Therefore we have proved .
Hello,
Sorry if my questions seem elementary, I am new to real analysis. Anyway I am having trouble with the following:
1. Prove that the empty set is unique. That is, suppose A and B are empty sets and prove A=B.
2. Prove, If U=A(union)B and A(intersection)B = empty set, then A=U\B.
Thank you for your help.
k
In general, to prove A= B for sets you prove and . And to prove you start with "if " and, using properties of A and B, conclude " ".
In this case, if then = A Union B and x is not in B since A intersect B is empty. Those together say U\B so U\B.
If U\B then but x is not in B. Since every member of of U is in either A or B, x must be in A proving U\B .
1) Suppose we have two empty sets Φ and Φ',if we can prove Φ=Φ' ,we are done.
So we know that the empty set is the subset of any other set X,thus:
................ ....for all X...................
IN this relation put X=Φ' And we get : .................................................. .....1
Since also Φ' is an empty set we have:
................. for all X...................................
And in this relation put X=Φ and we get : .................................................. .......................................2
Thus by (1) and (2) and equality of sets we get Φ=Φ'.......................