Take for instance the set \$X=a, b\$. I don"t check out \$emptyset\$ everywhere in \$X\$, so how can it it is in a subset? \$egingroup\$ "Subset of" means something different than "element of". Note \$a\$ is additionally a subset the \$X\$, regardless of \$ a \$ not showing up "in" \$X\$. \$endgroup\$
Because every solitary element the \$emptyset\$ is likewise an aspect of \$X\$. Or can you name an element of \$emptyset\$ that is not an element of \$X\$? that"s because there room statements that room vacuously true. \$Ysubseteq X\$ way for every \$yin Y\$, we have \$yin X\$. Currently is that true the for all \$yin emptyset \$, we have \$yin X\$? Yes, the statement is vacuously true, since you can"t pick any type of \$yinemptyset\$.

You are watching: Is empty set a subset of every set You have to start from the definition :

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

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

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

Now you have to use the truth-table definition of \$ ightarrow\$ ; you have actually that :

"if \$p\$ is false, climate \$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 above truth-definition that \$ ightarrow\$ gives us the :

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

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

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edited Jun 25 "19 at 13:51
answered january 29 "14 in ~ 21:55 Mauro ALLEGRANZAMauro ALLEGRANZA
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Subsets room not have to elements. The aspects of \$a,b\$ are \$a\$ and also \$b\$. Yet \$in\$ and also \$subseteq\$ are various things.

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answered january 29 "14 in ~ 19:04 Asaf Karagila♦Asaf Karagila
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