Take for instance the collection $X=\a, b\$. I don"t see $\emptyset$ anywhere in $X$, so how deserve to it it is in a subset? $\begingroup$ "Subset of" means something various than "element of". Note $\a\$ is additionally a subset that $X$, regardless of $\ a \$ not showing up "in" $X$. $\endgroup$
Because every solitary element that $\emptyset$ is additionally an aspect of $X$. Or have the right to you surname an element of $\emptyset$ the is no an aspect of $X$? that"s since there room statements that are vacuously true. $Y\subseteq X$ method for every $y\in Y$, we have actually $y\in X$. Now is that true the for all $y\in \emptyset$, we have $y\in X$? Yes, the statement is vacuously true, since you can"t pick any type of $y\in\emptyset$.

You are watching: Which set is a subset of every set? You need to start native the meaning :

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

Then friend "check" this an interpretation with $\emptyset$ in place of $Y$ :

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

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

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

so, because of the truth that :

$x \in \emptyset$

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

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

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

See more: Biology Chap 2 Compare And Contrast Habitat And Niche (With Comparison Chart)

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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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Subsets are not necessarily elements. The aspects of $\a,b\$ are $a$ and $b$. But $\in$ and $\subseteq$ are various things.

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