Take for instance the collection $X=a, b$. I don"t view $emptyset$ everywhere in $X$, so exactly how deserve to it be a subset?


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$egingroup$ "Subset of" implies something different than "element of". Note $a$ is additionally a subset of $X$, despite $ a $ not showing up "in" $X$. $endgroup$
that"s because tright here are statements that are vacuously true. $Ysubseteq X$ means for all $yin Y$, we have $yin X$. Now is it true that for all $yin emptyset $, we have $yin X$? Yes, the statement is vacuously true, because you can"t pick any $yinemptyset$.

You are watching: Is the empty set a subset of all sets


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Due to the fact that eincredibly single aspect of $emptyset$ is additionally an element of $X$. Or can you name an element of $emptyset$ that is not an element of $X$?


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You should start from the definition :

$Y subseteq X$ iff $forall x (x in Y ightarrow x in X)$.

Then you "check" this definition through $emptyset$ in area of $Y$ :

$emptyset subseteq X$ iff $forall x (x in emptycollection ightarrow x in X)$.

Now you need to usage the truth-table meaning of $ ightarrow$ ; you have actually that :

"if $p$ is false, then $p ightarrowhead q$ is true", for $q$ whatever;

so, due to the truth that :

$x in emptyset$

is not true, for eincredibly $x$, the over truth-interpretation of $ ightarrow$ gives us that :

"for all $x$, $x in emptyset ightarrowhead x in X$ is true", for $X$ whatever before.

This is the factor why the emptyset ($emptyset$) is a subset of every collection $X$.

See more: Bromine Is One Of Only Two Elements That Is A Liquid At Room Temperature


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edited Jun 25 "19 at 13:51
answered Jan 29 "14 at 21:55
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Mauro ALLEGRANZAMauro ALLEGRANZA
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Subsets are not necessarily facets. The aspects of $a,b$ are $a$ and $b$. But $in$ and also $subseteq$ are different points.


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answered Jan 29 "14 at 19:04
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Asaf Karagila♦Asaf Karagila
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