Are the following statements true? “{∅} = ∅” ? “{∅} ⊃ ∅”?

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

• Have you tried working through the definitions of all of the symbols involved? – Hurkyl Oct 7 '13 at 7:54
• Have you tried searching the site for something like this? – Asaf Karagila Oct 7 '13 at 7:59
• Yes I have. I couldn't find it. It's straight forward I was just curious for my exam tomorrow. – Jake Park Oct 7 '13 at 8:06
• 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 7 '13 at 8:31
• 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 7 '13 at 8:44

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 ∅ $\subset$ {∅}??

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 $\{\emptyset\}$?
• What's the difference between { } and no brackets? – Jake Park Oct 7 '13 at 7:59
• @JakePark Putting $\{\,\}$ round an element $A$, so writing $\{A\}$, means the set whose sole element is $A$. – Henry Oct 7 '13 at 8:58

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.