Notation for distinct members

What set theory notation should we use for something like “this set has n members that are apples and m members that are oranges”?

A clean way is to use cardinalities of subsets (or of property-defined subsets).

Let \(S\) be your set and let “\(x\) is an apple/orange” be predicates. Then write
\[ \bigl|{x\in S:\; x\text{ is an apple}}\bigr|=n, \qquad \bigl|{x\in S:\; x\text{ is an orange}}\bigr|=m. \]

Common equivalent formulations:

• If (A) and (O) are the (global) sets of apples and oranges,
  \(|S\cap A|=n\) and \(|S\cap O|=m\).
• If you label elements by a type map \(\tau:S\to{\text{apple},\text{orange}}\),
  \(|\tau^{-1}({\text{apple}})|=n\) and \(|\tau^{-1}({\text{orange}})|=m\).
• With indicator functions,
  \(\sum_{x\in S}\mathbf 1_{\text{apple}}(x)=n\) and \(\sum_{x\in S}\mathbf 1_{\text{orange}}(x)=m\).

If apples and oranges are mutually exclusive categories, you can also note \(A\cap O=\varnothing\); if those are the only categories in \(S\), then \(n+m=|S|\).