Take for instance the collection \$X=a, b\$. Ns don"t watch \$emptyset\$ everywhere in \$X\$, for this reason how can it it is in a subset? \$egingroup\$ "Subset of" method something various than "element of". Keep in mind \$a\$ is also a subset of \$X\$, in spite of \$ a \$ not appearing "in" \$X\$. \$endgroup\$
Because every solitary element that \$emptyset\$ is also an facet of \$X\$. Or deserve to you name an facet of \$emptyset\$ that is no an aspect of \$X\$? that"s because there space statements that are vacuously true. \$Ysubseteq X\$ means for all \$yin Y\$, we have actually \$yin X\$. Now is that true the for all \$yin emptyset \$, we have actually \$yin X\$? Yes, the declare is vacuously true, due to the fact that you can"t pick any type of \$yinemptyset\$.

You are watching: Is the empty set a subset of every set You need to start native the meaning :

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

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

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

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

"if \$p\$ is false, then \$p ightarrow q\$ is true", for \$q\$ whatever;

so, as result of the reality 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", for \$X\$ whatever.

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

See more: What Is The Oxidation State Of S In S2O32, What Is The Oxidation Number Of Sulfur In S2O3 2

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edited Jun 25 "19 in ~ 13:51
answered january 29 "14 in ~ 21:55 Mauro ALLEGRANZAMauro ALLEGRANZA
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Subsets space not necessarily elements. The elements of \$a,b\$ are \$a\$ and \$b\$. But \$in\$ and \$subseteq\$ are various things.

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