Take for example the set \$X=a, b\$. I don"t view \$emptyset\$ anywhere in \$X\$, so how can it be a subset?

\$egingroup\$ "Subset of" way something various than "element of". Keep in mind \$a\$ is likewise a subset the \$X\$, despite \$ a \$ not showing up "in" \$X\$. \$endgroup\$
Because every solitary element the \$emptyset\$ is likewise an element of \$X\$. Or deserve to you surname an facet of \$emptyset\$ that is not an aspect of \$X\$?

that"s since there are statements that space vacuously true. \$Ysubseteq X\$ method for every \$yin Y\$, we have actually \$yin X\$. Now is that true the for every \$yin emptyset \$, we have \$yin X\$? Yes, the statement is vacuously true, because you can"t pick any \$yinemptyset\$.

You are watching: Empty set is a subset of every set

You should start indigenous the an interpretation :

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

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

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

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

"if \$p\$ is false, then \$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 over truth-definition of \$ ightarrow\$ offers us that :

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

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

See more: What Is The Ongoing Process Of Tearing Down And Rebuilding Bone Matrix Is Called

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edited Jun 25 "19 in ~ 13:51
answered january 29 "14 at 21:55

Mauro ALLEGRANZAMauro ALLEGRANZA
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\$egingroup\$
Subsets room not necessarily elements. The elements of \$a,b\$ room \$a\$ and \$b\$. Yet \$in\$ and \$subseteq\$ are various things.

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answered jan 29 "14 in ~ 19:04

Asaf Karagila♦Asaf Karagila
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