Thread: Proving Basic Set Operations

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

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 .

Oh yes, that is so simple that it went right over my head initially. I see exactly what you mean!

Thank you!

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 .

Thank you again!

That makes sense now that I follow your logic. But I am having so much trouble constructing these from scratch! I will keep working at it, maybe it will start coming to me, like magic... ok, that is wishful thinking.

K

Originally Posted by math-chef

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

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'.......................