How do you write "The sum of all the elements in set A"?

Question

I was in class and the teacher started talking about the mean average. That got me thinking: how would I mathematically represent this? The worded defenition of the mean of set A is "the sum of all elements in set A divided by the amount of elements in set A". I started off with the "amount of elements in set A" bit, which I represented as n(A), though |A| could also work. But I got stuck when it came to the "sum of elements in set A" bit. How would I mathematically represent this?

Answer 1

What about the following notation: $$\sum_{a\in A}a$$

Answer 2

if the summation operation is defined, then the sum of all the elements in the set can be defined with the set-inclusion relation as follows

$$\sum_{a\in A} a$$

Answer 3

As long as we know that there is a way to add up elements of $A$, then writing $\sum_{a \in A} a$ is perfectly acceptable, as pointed out in the comments—but if you want to have an actual routine (or algorithm) for performing this sum, it would be valuable to have some sort of indexing on your set $A$.

Since you seem to be assuming that $A$ is finite, as you are talking about the cardinality/size $|A|$ of $A$, then it might be natural to think about the set $A$ as an indexed set, meaning that we attach a label to every element which tells us "which" element of $A$ we are looking at. So, if $|A| = n$, we might write $A = \{a_1,a_2,a_3,\dots,a_n\}$.

Now if you want to add up all the elements, this can be written as $\sum_{i = 1}^{n} a_i$. This is the same sum as $\sum_{a \in A} a$, but might fit more naturally with some notation you have seen in previous courses.