I know that if $a$ is an element of the set $A$ then this is denoted as $a\in A$. This is for all elements $a$ and for all sets $A$. If, however, $a$ is not an element of $A$, then this is denoted as $a\notin A$. The same applies if a set $B$ is contained within $A$, denoted as $B\subset A$ or $A\supset B$. And if the contrary, then $B\not\subset A$ or $A\not\supset B$.
If we have the set of elements $\{x, y, z\}$ that are common to both $A$ and $B$, then this is denoted as $\{x, y, z\} = A\cap B$. If the contrary, then this is denoted as $A\Delta B$. If we want to combine both sets together to form a new set $C$ with all of its elements either members of $A$ or $B$, then this is denoted as $C = A\cup B$.
What if $(*)$ $A$ had $n$ elements and $m$ of its elements were also in $B$ for some $0 < m < n$ then how do we write this? How would we write $(**)$ the contrary? By $(**)$, I mean how to denote if $A$ had $n$ elements and $m$ of its elements were not in $B$. I know that if $m = n$ then $(*)$ is denoted as $A = B$ or $A\leftrightarrow B$ or $A\subset B$ depending on the value of $m$, what elements are in $A$ and $B$, and how many elements are in $B$. If on the other hand $m = 0$, then this is denoted as $A\cap B = \varnothing =$ the empty (null) set. This makes $A$ and $B$ disjoint sets.
But if $0<m<n$ then how do we denote $(*)$ and $(**)$ using set notation? Would we just write $\{\ldots\} \subset B$ of some sort? I am not aware of a possible duplicate of this question.
Thank you in advance.
(*) $A$ has $n$ elements and $m<n$ of them are in $B$: $|A\cap B| = m$.
(**) $A$ has $n$ elements and $m<n$ of them are NOT in $B$: $|A\setminus B| = m$.
If you meant something else by "the contrary", please let me know!
EDIT: If, by "the contrary", you meant that $m<n$ elements of $A$ are in $A\Delta B$, then $|A\Delta B| \geq m$. You can write $A\setminus B \subset A\Delta B$ with $|A\setminus B| = m$. Or, if you mean that there are $m$ elements total in $A\Delta B$, this can obviously be written $|A\Delta B| = m$.