$egingroup$ "Subcollection of" implies somepoint different than "aspect of". Note $a$ is additionally a subset of $X$, despite $ a $ not appearing "in" $X$. $endgroup$
that"s bereason there 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, given that you can"t pick any type of $yinemptyset$.
You are watching: Why is the empty set a subset of every set
Since eextremely single facet of $emptyset$ is additionally an facet of $X$. Or can you name an aspect of $emptyset$ that is not an element of $X$?
You must start from the interpretation :
$Y subseteq X$ iff $forall x (x in Y ightarrowhead x in X)$.
Then you "check" this definition with $emptyset$ in location of $Y$ :
$emptyset subseteq X$ iff $forall x (x in emptycollection ightarrow x in X)$.
Now you must use the truth-table interpretation of $ ightarrow$ ; you have 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-definition of $ ightarrow$ offers us that :
"for all $x$, $x in emptycollection ightarrow x in X$ is true", for $X$ whatever before.
This is the factor why the emptyset ($emptyset$) is a subset of eextremely collection $X$.
See more: Why Does My Guinea Pig Drink So Much Water ? Why Is My Guinea Pig Drinking A Lot Of Water
edited Jun 25 "19 at 13:51
answered Jan 29 "14 at 21:55
Mauro ALLEGRANZAMauro ALLEGRANZA
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Subsets are not necessarily elements. The facets of $a,b$ are $a$ and also $b$. But $in$ and also $subseteq$ are different things.
answered Jan 29 "14 at 19:04
Asaf Karagila♦Asaf Karagila
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