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?

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

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?