Differing definitions for 'Algebra of subsets'

Question

For a collection, $A$ of subsets of a set $X$ to be an algebra of subsets it must satisfy the following properties:

1. $A$ is non-empty
2. If $E \in A \implies E^c \in A$
3. If $E, F \in A \implies E \cup F \in A$

However in other sources I have seen an algebra of subsets presented as:

1. $\emptyset, X \in A$
2. If $E \in A \implies E^c \in A$
3. If $E, F \in A \implies E \cup F \in A$

The first property in each definition does not seem equivalent to me. So is the definition of an algebra of subsets non-standard, i.e. it varies depending on the author of the text?

Answer 1

Well, if $E\in A$ then $E^c\in A$ so their union which is $X$ is in $A$ and and so is their intersection which is $\varnothing$.

So being non-empty is equivalent to having $X$ and $\varnothing$ in $A$.