The equality of three sets

Question

Let $A = \{ \emptyset \}$ , $B = \{ \emptyset , \emptyset\} $ and $C = \{ \emptyset , \{\}\}$ . Is $A = B = C$ right ? I think it's true because we can put $\{\}$ instead of $\emptyset$ and also remove duplicate elements .

Answer 1

Yes. The ultimate issue is notation and convention.

• $A=B$ because, generally, we ignore duplicate elements of sets. So, for instance, $\{1,1,1,1\}=\{1\}$.
• $B=C$ more obviously because both $\emptyset$ and $\{\}$ are notations for the empty set, so they're just representing the same thing in different ways.
• So clearly, $A=B=C$.

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Mostly posting this so this question can be finally considered to have an answer and thus be removed from the unanswered queue. Made it Community Wiki since I have nothing much of substance to add.