is the correlation between two things separated by a comma a compact way of writing two statements?

Question

I want to know if

corr({a,b},c) = 0

is the same as saying that

corr(a,c) = 0

corr(b,c) = 0

But that the first is just a short-hand way of writing it.

Answer 1

What is common is applying function to sets.
So if we have sets $M,N$, then $$corr(M,N) = \{ corr(m,n) \mid m \in M, n \in N\},$$

meaning that we apply the function to all elements. You would still need to define this term before using it, but no one should be surprised by it.

Now for your special case, you want $M = \{a,b\}$, $N = \{c\}$ and $corr(M,N) = \{0\}$. You could even define that $corr(M,N) = 0$ means that every element in $corr(M,N)$ is zero, this is also a common occurrence to save parentheses (e.g. when discussing vector spaces).