# Why is the empty set a subset of every set? [duplicate]

Take for example the set $X=\{a, b\}$. I don't see $\emptyset$ anywhere in $X$, so how can it be a subset?

• "Subset of" means something different than "element of". Note $\{a\}$ is also a subset of $X$, despite $\{ a \}$ not appearing "in" $X$. – user14972 Jan 29 '14 at 20:10
• Why does this question get a downvote? It is not hard to see that someone can be asking this seriously. – N. Owad Jan 29 '14 at 22:02
• Hint: Every element of the empty set is a pink elephant. Or an element of $X.$ (No joke) – Dan Christensen Jan 30 '14 at 4:59
• I personally like @HagenvonEitzen's question Or Can you name an element of ∅ that is not an element of X?. If you think it like this, given ∅ and X, you can't really find an element of ∅ (nothing) that you don't find in X, and as a subset A is a just a set whose elements (every element) are included in another set B, that is you can't find an element in A which is not in B, it makes more sense. – user3019105 Mar 15 '18 at 20:24

that's because there are statements that are vacuously true. $Y\subseteq X$ means for all $y\in Y$, we have $y\in X$. Now is it true that for all $y\in \emptyset$, we have $y\in X$? Yes, the statement is vacuously true, since you can't pick any $y\in\emptyset$.

Because every single element of $\emptyset$ is also an element of $X$. Or can you name an element of $\emptyset$ that is not an element of $X$?

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 $$X$$.

• Shouldn't the last implication be "$\text{for all x, }x \in \emptyset \rightarrow x \in X$ is true" – mauna Jun 30 '14 at 13:20

Subsets are not necessarily elements. The elements of $\{a,b\}$ are $a$ and $b$. But $\in$ and $\subseteq$ are different things.