What is the difference between these two subsets in Discrete Math?

Question

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 $U$?

Thanks

Answer 1

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 $\{\{1,2,3,4\}\}$.

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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 $1,2,3,4$.