For any set (nonempty or otherwise) $A, \varnothing \subseteq A$.
Two sets are said to be disjoint if they contain no elements in common, so for any nonempty set $A, \varnothing$ is disjoint from $A$.
Can the null set therefore be considered BOTH a subset of and disjoint from $A$? Doesn't that contradict the definitions of these terms?
Best Answer
Going straight to the definition: $A\subseteq B$ if whenever $a\in A$ we must have that $a\in B$. Well if $A=\emptyset$ this is true for every $B$ since there is no $a\in A$. Hence $\emptyset$ is a subset of every set.
$A$ and $B$ are disjoint if $A\cap B=\emptyset$. Clearly if $B=\emptyset$ then this is true for all $A$.
It may conflict with the English definition of the words, but math is not English. Words mean different things.