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I must prove the following:

Prop.: $(\emptyset,\{\emptyset\})$ is algebra of sets

Proof:

$\emptyset \in \{\emptyset\} $ by hypothesis

$\emptyset -\emptyset=\emptyset$ and by hypothesis $ \emptyset \in \{\emptyset\}$

$\emptyset \cup \emptyset= \emptyset$ and by hypothesis $ \emptyset \in \{\emptyset\}$

Therefore $(\emptyset,\{\emptyset\})$ is an algebra of sets.. Is it correct?

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mle
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  • This is incomplete. You also, at least, need $\varnothing\cup{\varnothing} = {\varnothing}\in A$, ${\varnothing} - \varnothing = {\varnothing}\in A$ (where $A$ is the purported algebra), right? And also ${\varnothing}\cup{\varnothing} = {\varnothing} \in A$, I suppose. – MPW Jun 06 '14 at 14:35
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    @MPW Based on another question I've seen the user post, I think s/he's using the notation $(M,A)$ to mean "$A\subseteq \mathcal{P}(M)$ is an algebra of subsets of $M$." – rschwieb Jun 06 '14 at 14:39
  • @rschwieb: Ah, I see. My bad. – MPW Jun 06 '14 at 14:41
  • @MPW np! If I hadn't seen that question I would have probably thought the same thing. These comments will probably prevent anyone else from thinking the same thing :) – rschwieb Jun 06 '14 at 14:43

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Yes, $\{\emptyset\}$ is an algebra of subsets of the set $\emptyset$. (Actually it's the only nonempty collection of subsets possible. :) )

You've shown the algebra contains the emptyset, is closed under complements and unions, and therefore intersections too.

rschwieb
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  • This is not correct, is it? The collections ${\varnothing}$ and ${\varnothing,{\varnothing}}$ are not the same. – MPW Jun 06 '14 at 14:38
  • @MPW Hi, please see my comment about the OP's choice of notation above. I don't believe he's claiming that ${\emptyset,{\emptyset}}$ is an algebra. – rschwieb Jun 06 '14 at 14:41
  • Understood. Thanks for pointing out the correct interpretation of the notation in this case. You're absolutely correct. Consider my previous comment(s) retracted. – MPW Jun 06 '14 at 14:42
  • @rschwieb.. thanks for answer! :) – mle Jun 06 '14 at 14:43
  • @rschwieb, in generally $(A,{A,\emptyset})$ is algebra.. is correct? – mle Jun 09 '14 at 22:33