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Are the following statements true? 
“{∅} = ∅” ?
 “{∅} ⊃ ∅” ? Stumbled upon this question. Was wondering what the answer was. Could you guys explain in return?

Jake Park
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    Have you tried working through the definitions of all of the symbols involved? –  Oct 07 '13 at 07:54
  • Have you tried searching the site for something like this? – Asaf Karagila Oct 07 '13 at 07:59
  • Yes I have. I couldn't find it. It's straight forward I was just curious for my exam tomorrow. – Jake Park Oct 07 '13 at 08:06
  • If you have an exam, and you can't answer that on your own, then I'm sorry to say that... but it's unlikely that you're going to pass. – Asaf Karagila Oct 07 '13 at 08:10
  • Also related: http://math.stackexchange.com/questions/202782/in-naive-set-theory and http://math.stackexchange.com/questions/491465/is-emptyset-a-subset-of-emptyset – Asaf Karagila Oct 07 '13 at 08:12
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    i feel the same way. i'm studying as much as i can. i'm finding this entire course to be challenging. – Jake Park Oct 07 '13 at 08:31
  • Then perhaps you've been studying wrong. My experience is that a lot of students just try to solve problems instead of understand definitions; whereas in mathematics (and logic and set theory in particular) solving problems on its own takes you nowhere. Understanding the definitions takes you all the way, without any additional efforts. – Asaf Karagila Oct 07 '13 at 08:37
  • my exma is on induction, sets and quantifiers. should i be watching more videos and rereading the textbook to help study better? i'm solving practice questions at this point in time. – Jake Park Oct 07 '13 at 08:38
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    You should solve problems in order to understand which definitions you don't understand. In this case, empty set, set equality, set inclusion. Review these definitions, solve this exercise on your own; continue by solving problems, if you don't understand a definition then go back to that, understand that, then solve the problem. That is how you study to an exam in mathematics. – Asaf Karagila Oct 07 '13 at 08:44

3 Answers3

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What is, e.g., {1}? The set whose sole member is 1 (the 'singleton' of 1).

Similarly, what is {∅}? The set whose sole member is ∅, i.e. whose sole member is the empty set.

So how many members does {∅} have? How many members does ∅ have?

What does $A \subset B$ mean? Do you see that that comes to: nothing is a member of $A$ and not a member of $B$? What if indeed nothing is a member of $A$?

OK, with those warm-ups, now ask yourself again: can {∅} be identical to ∅? Is it the case that ∅ $\subset$ {∅}??

Peter Smith
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Hint:

  • How many elements of $\emptyset$?
  • How many elements of $\{\emptyset\}$?
  • Are they the same?
  • Is every element of $\emptyset$ also an element of $\{\emptyset\}$?
  • Is any element of $\emptyset$ not an element of $\{\emptyset\}$?
  • Is $\emptyset$ a subset of $\{\emptyset\}$?
Henry
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Some of these might be more clear if you write them out in words, rather than symbols. For example, $$\{ \emptyset \} = \emptyset$$ is equivalent to saying

The set containing the empty set is equal to the empty set

This technique, combined with the questions posed by Henry, should bring you to your answers.

Aaron Taylor
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