Take for instance the collection $X=a, b$. Ns don"t watch $emptyset$ everywhere in $X$, for this reason how can it it is in a subset?


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$egingroup$ "Subset of" method something various than "element of". Keep in mind $a$ is also a subset of $X$, in spite of $ a $ not appearing "in" $X$. $endgroup$
Because every solitary element that $emptyset$ is also an facet of $X$. Or deserve to you name an facet of $emptyset$ that is no an aspect of $X$?


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that"s because there space statements that are vacuously true. $Ysubseteq X$ means for all $yin Y$, we have actually $yin X$. Now is that true the for all $yin emptyset $, we have actually $yin X$? Yes, the declare is vacuously true, due to the fact that you can"t pick any type of $yinemptyset$.

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


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You need to start native the meaning :

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

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

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

Now you should use the truth-table meaning of $ ightarrow$ ; you have actually that :

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

so, as result of the reality that :

$x in emptyset$

is not true, because that every $x$, the over truth-definition of $ ightarrow$ offers us that :

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

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

See more: What Is The Oxidation State Of S In S2O32, What Is The Oxidation Number Of Sulfur In S2O3 2


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edited Jun 25 "19 in ~ 13:51
answered january 29 "14 in ~ 21:55
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Mauro ALLEGRANZAMauro ALLEGRANZA
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Subsets space not necessarily elements. The elements of $a,b$ are $a$ and $b$. But $in$ and $subseteq$ are various things.


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