# Thread: Proving Basic Set Operations

1. ## 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.

k

2. The first one is simple logic.
The statment that "if $\displaystyle \color{blue}x \in \emptyset$ then $\displaystyle \color{blue}1=2$" is TRUE.
Recall that a false statement implies any statement and $\displaystyle x \in \emptyset$ is false.
So both of these are true:
$\displaystyle x \in \emptyset=A$ then $\displaystyle x \in B$ and $\displaystyle x \in \emptyset=B$ then $\displaystyle x \in A$
Therefore we have proved $\displaystyle A=B$.

3. ## Thank you

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

Thank you!

4. In general, to prove A= B for sets you prove $\displaystyle A\subset B$ and $\displaystyle B\subset A$. And to prove $\displaystyle A\subset B$ you start with "if $\displaystyle x\in A$" and, using properties of A and B, conclude "$\displaystyle x\in B$".

In this case, if $\displaystyle x\in A$ then $\displaystyle x\in U$= A Union B and x is not in B since A intersect B is empty. Those together say $\displaystyle x\in$ U\B so $\displaystyle A\subset$U\B.

If $\displaystyle x\in$ U\B then $\displaystyle x\in U$ 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$\displaystyle \subset A$.

5. ## I see

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

6. 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.

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:

................$\displaystyle \emptyset\subseteq X$ ....for all X...................

IN this relation put X=Φ' And we get : $\displaystyle \emptyset\subseteq\emptyset'$.................................................. .....1

Since also Φ' is an empty set we have:

.................$\displaystyle \emptyset'\subseteq X$ for all X...................................

And in this relation put X=Φ and we get :$\displaystyle \emptyset'\subseteq\emptyset$.................................................. .......................................2

Thus by (1) and (2) and equality of sets we get Φ=Φ'.......................