# If the complement of the universal set is the null set, then the null set is not in the universal set?

Let $\Omega$ be the universal set (which contains all objects of interest) and let its complement be denoted by $\Omega^c$ . The book I'm reading states that $\Omega^c = \phi$ ; does that mean $\phi \not\subset \Omega$?

• can you provide the book references please? – DJJ Sep 18 '18 at 18:06

Given the subset $A \subset \Omega$

The relative complement $A^c$ in $\Omega$ is defined as the set of elements of $\Omega$ that are not elements of $A$.

So in the case that $A = \Omega$, the relative complement of $A$ in $\Omega$ is equal to $\emptyset$.

That is, for every set $x$

we have $x \notin A^c$.

In particular we have $\emptyset \notin A^c$.

However, this is probably the "wrong" question to ask, because $\emptyset$ might not be an element of $\Omega$ at all.

Notice that the $\emptyset$ is a subset of every set, so in particular $\emptyset \subset \Omega$.

This is because to check whether $X \subset Y$ you take an arbitrary element of $X$ and check that it is also in $Y$. If $X$ is empty, then there is nothing to check.