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:

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


$$A \cap C = \emptyset$$

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

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


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