It suffices to show that (A') and (B') are equivalent. Exactly one of formula_10 or formula_11 is odd if and only if formula_17 is odd. If formula_18, formula_19 holds; hence, omission of symbols with the property formula_18 from a collection will not change the parity of formula_21. On the other hand, when formula_9, we have formula_23, but formula_24 simply enumerates the number of components with the property formula_9. Therefore, formula_26. Thus, formula_21 and the number of symbols in a collection with the property formula_9 have the same parity. Since formulations (A') and (B') are equivalent, so are (A) and (B), as claimed. □