Take for instance the set $X=a, b$. I don"t check out $emptyset$ everywhere in $X$, so how can it it is in a subset?


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$egingroup$ "Subset of" means something different than "element of". Note $a$ is additionally a subset the $X$, regardless of $ a $ not showing up "in" $X$. $endgroup$
Because every solitary element the $emptyset$ is likewise an aspect of $X$. Or can you name an element of $emptyset$ that is not an element of $X$?


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that"s because there room statements that room vacuously true. $Ysubseteq X$ way for every $yin Y$, we have $yin X$. Currently is that true the for all $yin emptyset $, we have $yin X$? Yes, the statement is vacuously true, since you can"t pick any type of $yinemptyset$.

You are watching: Is empty set a subset of every set


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You have to start from the definition :

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

Then friend "check" this definition with $emptyset$ in location of $Y$ :

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

Now you have to use the truth-table definition of $ ightarrow$ ; you have actually that :

"if $p$ is false, climate $p ightarrow q$ is true", because that $q$ whatever;

so, due to the truth that :

$x in emptyset$

is not true, because that every $x$, the above truth-definition that $ ightarrow$ gives us the :

"for all $x$, $x in emptyset ightarrow x in X$ is true", because that $X$ whatever.

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

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edited Jun 25 "19 at 13:51
answered january 29 "14 in ~ 21:55
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Mauro ALLEGRANZAMauro ALLEGRANZA
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Subsets room not have to elements. The aspects of $a,b$ are $a$ and also $b$. Yet $in$ and also $subseteq$ are various things.


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answered january 29 "14 in ~ 19:04
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Asaf Karagila♦Asaf Karagila
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