# Thread: Sets and Venn diagrams

1. ## Sets and Venn diagrams

Hi,I was just wondering if anyone could help me with this question?Simplify:A∪A′ for any set A∈U

2. ## Re: Sets and Venn diagrams

Originally Posted by vanilla5085
Hi,I was just wondering if anyone could help me with this question?Simplify:A∪A′ for any set A∈U
For any $x\in\mathcal{U}$ it is true that $x\in A\text{ or }x\notin A$ so $A\cup A'=~?$

3. ## Re: Sets and Venn diagrams

U, by definition of A'.

4. ## Re: Sets and Venn diagrams

Thanks Plato and Hartlw,
the answer is U but I don't understand why.

5. ## Re: Sets and Venn diagrams

A U A' means "the element chosen could come from either set 'A' or set 'not A' or both". Since it's impossible to be from both, that means it has to either come from A or not come from it. Thus it could be anything in the universal set.

6. ## Re: Sets and Venn diagrams

A' is all elements in U not in A. If you add A you get U.

7. ## Re: Sets and Venn diagrams

First show that $A\cup A' \subseteq U$. Then show $U \subseteq A \cup A'$.

Claim: $A \cup A' \subseteq U$

Proof:
Since $A \subseteq U$ so $A' \subseteq U$. Hence, given any $x \in A \cup A'$, at least one of the following is true: $x \in A$ or $x \in A'$. In either case, by the definition of subset, $x \in U$, so $A \cup A' \subseteq U$ as claimed.

Claim: $U \subseteq A \cup A'$

Proof: See post #2 (from Plato) or #5 (from Prove It)

8. ## Re: Sets and Venn diagrams

Originally Posted by Hartlw
A' is all elements in U not in A. If you add A you get U.
That is perfectly clear and correct, based on definition of A'. The abstractions are redundant.

9. ## Re: Sets and Venn diagrams

Thanks everyone!