TITLE: Why is the empty set a subset of every set?
QUESTION [8 upvotes]: Take for example the set $X=\{a, b\}$. I don't see $\emptyset$ anywhere in $X$, so how can it be a subset?

REPLY [14 votes]: You must start from the definition :

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

Then you "check" this definition with $\emptyset$ in place of $Y$ :

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

Now you must use the truth-table definition of $\rightarrow$ ; you have 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 of $\rightarrow$ gives 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 set $X$.