Empty set

The empty set is the set $\varnothing$ such that

It is the identity for union in the sense that $A\cup \varnothing = A$, and it is the smallest set under $\subseteq$. In analysis, it often appears as a preimage of an impossible condition (e.g., $f^{-1}(B)=\varnothing$ when $B$ misses the range of $f$).

Examples:

• $\{x\in\mathbb{R}: x^2+1=0\}=\varnothing$.
• $(0,1)\cap (1,2)=\varnothing$.
• For any set $A$, $A\setminus A=\varnothing$.