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Let $x$ it is in a set. What is the difference in between $x$ and also $x$? I get that the latter is a collection consisting of a single element - namely $x$, however what is the difference?

For example, we deserve to have $x$ to be the set $1$, climate $x=\1\$. Aren"t those $2$ expression the same?

Another problem are the base - when we have actually a set, execute we always have to surround him with brackets, for instance, deserve to we have actually $x$ to be the collection $2$?

Thanks a lot


elementary-set-theory notation
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edited january 26 "18 in ~ 8:20
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Asaf Karagila♦
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asked january 25 "18 at 13:32
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asdfasdf
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$egingroup$
Think that the brackets as a bag you placed things in. Climate $1$ is a bag include the number $1$. Yet $\1\$ is a bag containing a bag containing the number $1$. So two bags, one inside the other. These room different. Physically various if friend think real paper bags.


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answered jan 25 "18 in ~ 13:38
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Ethan BolkerEthan Bolker
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$$1 $$ is a collection whose the unique aspect is the creature $1$

$$\1\ $$ is a set whose the unique aspect is the set $1 $.


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answered jan 25 "18 at 13:35
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hamam_Abdallahhamam_Abdallah
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You room probably gaining confused between the surname of a set and that is description.When we compose $A=x$, we average $A$ is a set and inside collection $A$, we have an aspect $x$.

Now if I define another set $B=A$, then $B$ is a set and inside collection $B$, we have actually an element $A$, i beg your pardon is additionally a set. In this case, $B$ is a set of sets.

If you want to refer to the last set, writeits surname $B$, orits summary $A$.

For your last question, YES, we surround the aspects of the collection by curly braces , which also ensures unorderdness and also non-repeatability.


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edited jan 25 "18 in ~ 19:39
answered jan 25 "18 in ~ 13:45
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spkakkarspkakkar
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Well if you have actually $x=varnothing$, then $0=#x eq #x=1$. So plainly both sets room not the same.

Edit: with $#S$ I describe the cardinality of a collection $S$, i.e. In the finite instance the number of elements in $S$.


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edited january 27 "18 in ~ 8:24
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asdqasdq
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There is a practical distinction when friend think about how you can use these sets - namely together a domain that functions. A role that bring away a number is no the same as a duty that bring away a set.


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answered jan 25 "18 in ~ 14:42
Kevin OlreeKevin Olree
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Perhaps it would be beneficial to imagine the distinction in concrete terms - speak in regards to a computer data structure. Expect we represent sets using connected lists . Then $x$ is (the head reminder of) a succession of nodes, where each node has actually a guideline to an element of $x$ and a pointer to the next node in the sequence. Climate $x$ is (the head tip of) a solitary node, whose element pointer points to (the head node of) the sequence of nodes representing $x$. Clearer?


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answered january 25 "18 at 15:17
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