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What is the difference between $\bigcup\left\{ \emptyset,\left\{ 1\right\} \right\}$ and $\bigcup\left\{ \emptyset, 1 \right\}$ ?

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    $\begingroup$ Why you have the union symbol? $\endgroup$ – MonsieurGalois Jan 19 '17 at 23:20
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    $\begingroup$ The first one is $\emptyset \cup \{ 1 \} = \{ 1\}$ , while the second one is $\emptyset \cup 1= 1$. $\endgroup$ – Crostul Jan 19 '17 at 23:21
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Remember that "$\bigcup X$" means "The set of all things which are elements of elements of $X$."

So:

  • If $X=\{\emptyset, \{1\}\}$: there are two elements of $X$; the first has no elements, the second has one element, namely "$1$"; so $1$ is the only element of an element of $X$. So $\bigcup X=\{1\}$.

  • If $X=\{\emptyset, 1\}$: again, the first element has no elements. So the answer is just "The set of all elements of $1$." This is a bit weird at first, but remember that in set theory, $1$ is a set - namely, $1=\{\emptyset\}$. So $\bigcup X=1$.

In general, if $a$ is a set then $\bigcup\{\emptyset, a\}=a$.

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This can be easily reduced to, what is the difference between 1 and {1}.

The first is a number, the second is a bag containing the number.

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