Take for instance the collection \$X=a, b\$. I don"t view \$emptyset\$ everywhere in \$X\$, so exactly how deserve to it be a subset? \$egingroup\$ "Subset of" implies something different than "element of". Note \$a\$ is additionally a subset of \$X\$, despite \$ a \$ not showing up "in" \$X\$. \$endgroup\$
that"s because tright here 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, because you can"t pick any \$yinemptyset\$.

You are watching: Is the empty set a subset of all sets Due to the fact that eincredibly single aspect of \$emptyset\$ is additionally an element of \$X\$. Or can you name an element of \$emptyset\$ that is not an element of \$X\$? You should start from the definition :

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

Then you "check" this definition through \$emptyset\$ in area of \$Y\$ :

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

Now you need to usage the truth-table meaning of \$ ightarrow\$ ; you have actually 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-interpretation of \$ ightarrow\$ gives us that :

"for all \$x\$, \$x in emptyset ightarrowhead x in X\$ is true", for \$X\$ whatever before.

This is the factor why the emptyset (\$emptyset\$) is a subset of every collection \$X\$.

See more: Bromine Is One Of Only Two Elements That Is A Liquid At Room Temperature

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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 facets. The aspects of \$a,b\$ are \$a\$ and \$b\$. But \$in\$ and also \$subseteq\$ are different points.

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