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

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- February 18th 2014, 01:46 AMvanilla5085Sets and Venn diagrams
Hi,I was just wondering if anyone could help me with this question?Simplify:A∪A′ for any set A∈U

- February 18th 2014, 02:20 AMPlatoRe: Sets and Venn diagrams
- February 18th 2014, 03:59 AMHartlwRe: Sets and Venn diagrams
U, by definition of A'.

- February 18th 2014, 09:37 PMvanilla5085Re: Sets and Venn diagrams
Thanks Plato and Hartlw,

the answer is U but I don't understand why. - February 18th 2014, 09:53 PMProve ItRe: 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.

- February 19th 2014, 05:12 AMHartlwRe: Sets and Venn diagrams
A' is all elements in U not in A. If you add A you get U.

- February 19th 2014, 05:29 AMSlipEternalRe: 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) - February 19th 2014, 05:44 AMHartlwRe: Sets and Venn diagrams
- February 21st 2014, 01:41 AMvanilla5085Re: Sets and Venn diagrams
Thanks everyone!