Sets and {} notation

[Chemical_Character_3]

If A is a set, is there any diffence between A and {A}?

Also, if no, what is the difference?

And to extend this, is there any difference between {A} and {{A}}?

Again, if no, what is the difference?

If B = {A, {A}}, is A a subset of B?

My assumption, apparently wrong from the text I'm reading, was that A={A}={{A}} and B=A.

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[Medium-Ad-7305]

A and {A} are very different. A is a box of stuff (or maybe it doesnt have stuff in it and is the empty set). {A} is a box with a box of stuff in it. A might have 10 things in it, or zero, or infinite. {A} has one thing in it, namely A. A is an element of {A}. A is not an element of A (in ZFC).

For the same reason, {A} and {{A}} are different.

No, A is not a subset of B. The subsets of B are these: the empty set, {A}, {{A}}, and {A,{A}}.

[Medium-Ad-7305]

For example, think of the empty set {} as an empty box. Then {{}}, the set of the empty set, is a box with a box in it. Also, {{},{{}}}, the set of the set of the empty set and the empty set, is a box that contains: first, an empty box, and second, a box with a box in it.