I need to prove the following letting A and B be subsets of a universal set U. $$A\cup B = U \iff A^c\subset B$$ It just isn't straight forward to me.
Proving set theory question
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elementary-set-theory
proof-writing
1 Answers
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Think about it element-by-element. To show that $A \cup B = U \implies A^c \subset B$, you want to assume $A \cup B = U$ and show that $A^c \subset B$. What "$A^c \subset B$" means is that every member of $A^c$ is in $B$. So let $x \in A^c$. Since $A \cup B = U$ and $x$ isn't in $A$, it must be in $B$. So $x \in B$. Since $x$ was an arbitrary member of $A^c$, we have that $A^c \subset B$.
I'll leave the other direction to you, but use the same sort of approach.