# If two sets $A$ and $C$ do not intersect is the insect A $\cap$ C nothing, or the set of nothing?

This is a basic question, but I am unsure of the answer. If I have sets $A$ and $C$, and the two sets do not intersect, would I notate it as:

$$A \cap C = \{\emptyset\}$$

or

$$A \cap C = \emptyset$$

I'm leaning towards the first, because $\emptyset$ is an element of all sets right?

• the latter, the empty set is not an element of all sets, but it is a subset of every set – mdave16 Oct 23 '17 at 1:13
• $\varnothing$ is a subset of every set. – Clive Newstead Oct 23 '17 at 1:13
• The insect is infintetesimal. – William Elliot Oct 23 '17 at 1:24
• Who else came here for the bug? – zwim Oct 23 '17 at 2:07

What it means for $A$ and $C$ to be disjoint, or to not intersect, is that they have no elements in common.
Spelling it out, this means that there does not exist $x$ such that $x \in A$ and $x \in C$. By definition of intersection, this says exactly that there does not exist $x$ such that $x \in A \cap C$. Thus, $A \cap C$ has no elements.
In summary: $$A \cap C = \varnothing$$ It is not the case that $\varnothing$ is an element of every set. It is a subset of every set, though!