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.

  • 2
    $\begingroup$ The good reason is that $K'$ is not the complement of $K$, but just another set. $\endgroup$ – J.-E. Pin Sep 28 '20 at 5:24
  • $\begingroup$ Oh I see! I shall read it again in a new light $\endgroup$ – Friedrich 'Fred' Clausen Sep 28 '20 at 5:25
  • $\begingroup$ To be more precise, $K$ and $K'$ are both subsets of some set $E$. $\endgroup$ – J.-E. Pin Sep 28 '20 at 5:25
  • $\begingroup$ @J.-E.Pin: Specifically, ‘$K$ and $K'$ are any sets of integers’ (from the sentence introducing the equation). $\endgroup$ – Brian M. Scott Sep 28 '20 at 5:36

As J.-E. Pin and Brian M. Scott pointed out in the comments this was due to me not properly reading the part where they say

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

Now it makes sense. I shall certainly endeavour to read more carefully in the future ☕


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