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Let $U = \{1, 2, 3, 4, 5, 6, x, y, \{1, 2\}, \{1, 2, 3, 4\}, \{1, 2, 3\}\}$

$A = \{1, 2, 3, 4\}$

Can anyone explain to me the difference between these 2 pairs of things?

$$A\subseteq U \text{ and } \{A\} \subseteq U $$

And how come $\{A\}$ is not an element of $U$?


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One is a question if A itself is a subset of U, i.e. all elements of A are in U, the other is as to if the set containing as its only element A is a subset of U, i.e. is $A\in U$? – Adam Hughes Jul 1 '14 at 2:36
@anorton: You messed up the braces on $\{A\}$, and I can't fix it or suggest the edit because of the minimum character limit on edits. – user2357112 Jul 1 '14 at 4:26
In line 5 I've changed $A$ back to $\{A\}$ which I think is what you want - please let me know if not. – David Jul 1 '14 at 5:12
up vote 10 down vote accepted

The statement $A\subseteq U$ means that every element of $A$ is an element of $U$: in this case, $1,2,3,4$ are all in $U$: true, they are the first four elements in your listing.

The statement $\{A\}\subseteq U$ means that every element of $\{A\}$ is in $U$: that is, $A$ itself (not the elements of $A$) is in $U$: that is, $\{1,2,3,4\}$ is in $U$. This is also true as $\{1,2,3,4\}$ is the second last element you have listed in $U$. Notice however that this statement is different from the previous one: to confirm that one was true you had to look at the first four elements of $U$, not the second last.

However, $\{A\}\in U$ means that the whole expression $\{A\}$ is an element of $U$: that is, $\{\{1,2,3,4\}\}$ is an element of $U$. This is false: if you look carefully at $U$ you will see that its elements are $$1,\,2,\,3,\,4,\,5,\,6,\,x,\,y,\,\{1,2\},\,\{1,2,3,4\},\,\{1,2,3\}\ ,$$ and none of these is identical with $\{\{1,2,3,4\}\}$.

If it helps, think of a set as a bag containing stuff. The bag $U$ contains the items $1,2,3,4$; it also contains a bag containing "duplicate copies of" $1,2,3,4$; but it does not contain a bag containing a bag containing $1,2,3,4$.

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thanks, i was mainly confused with the {A} part because i wasn't sure if it was {{1,2,3,4}} or just {1,2,3,4} since my Grimaldi textbook is vague and doens't explain. – user3718584 Jul 1 '14 at 2:56
Wait a minute, the way you described the first one for {A} is a subset U kinda sounds like the same thing as describing A is an element of U? – user3718584 Jul 1 '14 at 3:25
Yes, it is exactly the same... but not the same as saying $\{A\}$ is an element of $U$. – David Jul 1 '14 at 3:26

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