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Is the set {a,$\emptyset$} = {a} ?

My initial guess guess was that it should not make a difference between the two sets as we think of $\emptyset$ being a subset of every set. Hence I concluded that $\emptyset$ must belong to every set .

However, I am still unsure , any answer indicating the fallacy of my logic or correct answers will be appreciated.

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    $\begingroup$ The set $\{a\}$ has one element. The set $\{a, \emptyset\}$ has two. They are not equal. $\endgroup$ – user296602 Jul 5 '16 at 4:57
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    $\begingroup$ $\emptyset$ is contained in every set as a subset but it is not an element of every set. $\endgroup$ – JMoravitz Jul 5 '16 at 4:59
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    $\begingroup$ Are a bag containing an apple and a bag containing an apple as well as another bag identical things? $\endgroup$ – Zev Chonoles Jul 5 '16 at 5:01
  • $\begingroup$ @ZevChonoles OUTSTANDING observation/question,Zev! $\endgroup$ – Mathemagician1234 Jul 5 '16 at 5:35
  • $\begingroup$ Without a doubt, this is a duplicate. But I can't go hunting for it right now. $\endgroup$ – Asaf Karagila Jul 5 '16 at 6:03
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You are confusing two concepts - for set A to be a subset of set B, it means that any element of A is also in B. For example, $\{2, 4, 6\}$ is a subset of $\{1, 2, 3, 4, 5, 6\}$. As you note, the empty set $\emptyset = \{\}$ is a subset of every set, because any element of the empty set (of which there are none) is also an element of any other set you can name.

For an item A to be an element of set B, it means that somewhere in the collection of things that are in B, A is included. It is possible for a set to be an element of another set - for example, $\{2, 4, 6\}$ is an element of $\{1, 2, 3, \{2, 4, 6\}\}$, but it is not a subset because if you go through the elements of A individually, you will see that neither 4 nor 6 is an element of B. In general, $\emptyset$ is not an element of most** sets, because most** sets, if you were to list out their elements, would not include the empty set. For example, $\emptyset$ is not an element of $\{2, 4, 6\}$.

A related concept is that of the union of sets. The union of two sets is the set that contains all elements that are in at least one of the sets. For example, the union of $\{1, 2, 3, 4\}$ and $\{4, 5, 6\}$ is $\{1, 2, 3, 4, 5, 6\}$ (noting that the repeated element 4 only appears once in the final set). It is not hard to prove that for any set A, the union of A and $\emptyset$ is A, because "adding" the elements of $\emptyset$ to anything won't give you any new elements.

** For a given definition of "most".

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Consider what it means for 2 sets to be equal. It means they are subsets of each other-or to be more precise, every element of one is an element of the other. It's very important to understand that while another distinct set can be an element of a set, a subset cannot be an element of the universal set unless we're dealing with a well defined collection of sets,such as the power set of a set, which is the collection of all it's subsets.

Notice the power set does not have any single elements as it's members.This is also true of any well defined collection of sets in ZF set theory or any comparable set theory. When we discuss sets formally, this restriction is placed by the Axiom of Comprehension and it's postulated in order to avoid paradoxes like Russell's Paradox.

So let's look at the sets {a,{$\emptyset$}} and {a}. Clearly they are not the same since the first set contains the empty set as an element rather then as a subset. Another way to put it is that the first set has 2 elements and the second set has one. (We could do this more formally by showing a bijection cannot be established between the sets, but I don't think we have to be this formal with 2 such simple sets.)

As Zev Chonoles made the point brilliantly above in his comment, a bag containing an empty bag and an apple isn't the same as a bag containing an apple merely because the bag is empty!

(I'm totally gonna steal that one for my students.........lol )

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