# Concrete Mathematics: Clarification about equation 2.20: intersection of set with its complement

In Concrete Mathematics (Graham, Knuth, and Patashnik) equation 2.20 is

$$\sum_{k\in K}a_k + \sum_{k \in K'}a_k = \sum_{k \in K \cap K'}a_k + \sum_{k \in K \cup K'}a_k$$

Specifically $$k \in K \cap K'$$: I would have thought a set cannot interect with its complement? So the equation could just be

$$\sum_{k\in K}a_k + \sum_{k \in K'} = \sum_{k \in K \cup K'}a_k$$

I am sure the authors had a good reason to include the intersection RHS sum and I want to make sure I understand the point they're making.

• The good reason is that $K'$ is not the complement of $K$, but just another set. – J.-E. Pin Sep 28 '20 at 5:24
• Oh I see! I shall read it again in a new light – Friedrich 'Fred' Clausen Sep 28 '20 at 5:25
• To be more precise, $K$ and $K'$ are both subsets of some set $E$. – J.-E. Pin Sep 28 '20 at 5:25
• @J.-E.Pin: Specifically, ‘$K$ and $K'$ are any sets of integers’ (from the sentence introducing the equation). – Brian M. Scott Sep 28 '20 at 5:36

If $$K$$ and $$K'$$ are any sets of integers