Take for example the set \$X=a, b\$. I do not see \$emptyset\$ everywhere in \$X\$, so just how have the right to it be a subset? \$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

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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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Subsets are not necessarily elements. The facets of \$a,b\$ are \$a\$ and also \$b\$. But \$in\$ and also \$subseteq\$ are different things.

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