Parity and arity

17 Mar 2020 ⇐ Notes archive

(This is an entry in my technical notebook. There will likely be typos, mistakes, or wider logical leaps — the intent here is to “let others look over my shoulder while I figure things out.”)

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Two tucked-away, somewhat-related terms I enjoy: parity and arity.

The former is the odd or even-ness of an integer.

The latter describes the number of arguments a function accepts.

Example usage of parity:

Today I learned about the Handshaking Lemma. It states that any finite undirected graph, will have an even number of vertices with an odd degree.

The proof rests on parity. Specifically, if you sum the degrees of every vertex in a graph, you’ll double count each edge. And that double counting implies the sum is even, and even parity is only maintained if there is an even—including zero—number of vertices with an odd degree.

Put arithmetically, a sum can only be even if its components contain an even number of odd terms.

Examples of arity:

• Swift’s tuple comparison operators topping out at arity six.
• Publisher.zip is only overloaded to arity three.
  • I PR’d a variadic overload to CombineCommunity/CombineExt yesterday and have a post walking through it in the works.