Take for example the collection $X=\a, b\$. I don"t see $\emptyset$ anywhere in $X$, therefore how have the right to it it is in a subset?


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$\begingroup$ "Subset of" means something different than "element of". Keep in mind $\a\$ is additionally a subset that $X$, despite $\ a \$ not showing up "in" $X$. $\endgroup$
that"s since there room statements that space vacuously true. $Y\subseteq X$ means for every $y\in Y$, we have $y\in X$. Now is the true the for every $y\in \emptyset $, we have actually $y\in X$? Yes, the explain is vacuously true, since you can"t pick any $y\in\emptyset$.

You are watching: The empty set is a subset of every set


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Because every solitary element that $\emptyset$ is additionally an element of $X$. Or can you surname an element of $\emptyset$ that is no an aspect of $X$?


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You have to start native the definition :

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

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

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

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

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

so, due to the fact that :

$x \in \emptyset$

is not true, for every $x$, the above truth-definition that $\rightarrow$ provides us that :

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

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

See more: The Maize At Little Darby Creek In, The Maize At Little Darby Creek


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


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