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$A$ is not a subset of $B$ if $A$ contains an element that is not in $B$. $A$ and $B$ can still have elements in common. We write this $A\not\subset B$.
$$\Z\not\subseteq\N,\quad\Z\not\subset\Z,\quad \{\Tgreen{A}, \Tred{O}, \Tgreen{I}\}\not\subset\{\Tgreen{A}, B, \Tgreen{I}, L\}$$
The empty set has no elements. We write it $\emptyset$. It is a subset of all sets.