$egingroup$ "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

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 $X$?

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

Mauro ALLEGRANZAMauro ALLEGRANZA

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$endgroup$

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$egingroup$

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

Asaf Karagila♦Asaf Karagila

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$endgroup$

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