$egingroup$ "Subset of" way something various than "element of". Keep in mind $a$ is likewise a subset the $X$, despite $ a $ not showing up "in" $X$. $endgroup$
Because every solitary element the $emptyset$ is likewise an element of $X$. Or deserve to you surname an facet of $emptyset$ that is not an aspect of $X$?
that"s since there are statements that space vacuously true. $Ysubseteq X$ method for every $yin Y$, we have actually $yin X$. Now is that true the for every $yin emptyset $, we have $yin X$? Yes, the statement is vacuously true, because you can"t pick any $yinemptyset$.
You are watching: Empty set is a subset of every set
You should start indigenous the an interpretation :
$Y subseteq X$ iff $forall x (x in Y ightarrow x in X)$.
Then girlfriend "check" this meaning with $emptyset$ in location of $Y$ :
$emptyset subseteq X$ iff $forall x (x in emptyset ightarrow x in X)$.
Now you need to use the truth-table definition of $ ightarrow$ ; you have actually that :
"if $p$ is false, then $p ightarrow q$ is true", because that $q$ whatever;
so, due to the truth that :
$x in emptyset$
is not true, because that every $x$, the over truth-definition of $ ightarrow$ offers us that :
"for all $x$, $x in emptyset ightarrow x in X$ is true", because that $X$ whatever.
This is the factor why the emptyset ($emptyset$) is a subset the every set $X$.
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edited Jun 25 "19 in ~ 13:51
answered january 29 "14 at 21:55
Mauro ALLEGRANZAMauro ALLEGRANZA
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Subsets room not necessarily elements. The elements of $a,b$ room $a$ and $b$. Yet $in$ and $subseteq$ are various things.
answered jan 29 "14 in ~ 19:04
Asaf Karagila♦Asaf Karagila
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