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I am reading Topology by James Munkres and I got the following question:

I understand that a is an element and {a} is a set containing the element a. I can memorize it, but as I reason it I think that I do not have the difference entirely clear in my head.

This is because I thought that a set itself is only defined by the elements it is made of. That is to say, that the set itself it is nothing other than the elements it is made of. Now the only difference I perceive from a and {a} is that the later would content the elements of the empty set (since it is assumed that the empty set is a subset of all sets).

If that is not the only difference between both notions, how would you explain it? Again, I can memorize that they are not different and that visually they are not. But I think that I can understand deeper the concept of a set if I get the difference it has with an element very clear. Thanks.

José Carlos Santos
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César D. Vázquez
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  • $\emptyset$ is the empty set and has no elements. ${\emptyset}$ has one element and so is not the empty set. – Henry Feb 21 '20 at 12:23
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    I can give you a cookie, or a box containing a cookie. That's about the same difference between $a$ and ${a}$. – Suzet Feb 21 '20 at 12:39

1 Answers1

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If $a=\{1,2,3\}$ then the set $a$ has $3$ elements, whereas the set $\{a\}$ has a single element, which is $a$. So, $a\neq\{a\}$.

José Carlos Santos
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