How does one read this $$total = \sum_{i\in A}\left( C_0-C_i\right)+ \sum_{i\in B}\left( C_0-C_i\right)+ \sum_{i\in C}\left( C_0-C_i\right)$$ $$ = C_0\left(|A|+ |B|+ |C|\right)-\sum_{i\in A\cup B\cup C}C_i$$

The way I read this is total is equal to $C_0$ times the absolute values of A + B + C - sum of $C_i$ of A, B, or Ci if they exist? Is this right?

  • 4
    $\begingroup$ When $A$ is a set $|A|$ is its cardinality. Also, the given equality only holds if $A,B,C$ are mutually disjoint. $\endgroup$
    – dxiv
    Oct 15, 2016 at 0:15
  • 1
    $\begingroup$ @dxiv Seems like this should be an answer :) $\endgroup$
    – arkeet
    Oct 15, 2016 at 0:55

1 Answer 1


I can see how you can be confused because the choice of symbols in this expression is poor. They are using $C$ both for the indexed terms in sums ($Ci$) and for a set that the indexes come from. Let's dissect the expression.

I think what confuses you is $i\in A\cup B\cup C$.

$A\cup B\cup C$ is the set where the index of the sum takes its values. Let's take a simple example, where the set is just the three numbers 1,2,3. $$\sum_{i\in\{1,2,3\}}C_i = C_1 +C_2 +C_3$$ I hope this is clear to you.

Now in our case, $A, B, C$ are just sets (of natural numbers one would assume). For example $A$ could be $\{3,4,5\}$, $B$ could be $\{2,1\}$, and $C$ could be $\{3,4,10,20\}$. $A\cup B\cup C$ is the union of these sets, so for our example, it would be $\{1,2,3,4,5,10,20\}$

As pointed out in the comments if we have a set $S$, then $|S|$ is the cardinality of the set, i.e., how many elements the set has. So for our example above, the expression you give becomes:

$$ C_0\left(|A|+ |B|+ |C|\right)-\sum_{i\in A\cup B\cup C}C_i=$$ $$ C_0(3+2+4) - \sum_{i\in\{1,2,3,4,5,10,20\}}C_i = $$ $$ 9C_0 - (C_1+C_2+C_3+C_4+C_5+C_{10}+C_{20})$$

As pointed in the comments, the equation you gave only holds if the sets A, B, C are mutually disjoint (they do not share any elements). You can find out why by exploring the example I just gave you, where the sets are not mutually disjoint. Can you see why the equation does not hold?


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .