Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I need to prove $$A\cup B = \Omega \Longleftrightarrow{A^c \subseteq B}$$

How can I do it? This is what I've got so far, but I don't know if it is a valid demonstration:

$A\cup B = \Omega \Longleftrightarrow{x\in{A}\vee x\in{B}}\Longleftrightarrow{x\notin{A}\rightarrow{x\in{B}}}\Longleftrightarrow{A^c\subseteq B}$

Is this correct? If not, how can I prove that? Thanks in advanced.

share|improve this question
    
There is the problem of the missing quantifiers, identified by Arturo Magidin. Also, if you have any doubts, don't do things in one line. You need to prove two things, $A \cup B\rightarrow A^c\subseteq B$, and the implication that runs the other way. Prove them separately, you will retain better control of the logic. Now for the first direction. Suppose that $A\cup B=\Omega$. Let $x\in A^c$. So $x$ is not in $A$. Since $x\in A\cup B$, $x$ must be in $B$. I leave the other direction to you. –  André Nicolas Nov 15 '11 at 18:17
add comment

1 Answer

up vote 4 down vote accepted

The second clause should really be "For all $x$, $x\in A\lor x\in B$" (or $\forall x(x\in A\lor x\in B)$), and the quantifier should be repeated in the next one; otherwise, it looks correct.

(Except that you are proving, not proofing).

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.