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

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.

• I haven't seen it being used this way but if you explicitly define it I think it can make sense in certain contexts. But I don't think it is a standard notation. – flawr Apr 16 at 7:22
• I too think it looks weird. What is done is two regressions Y1 = B1*X1 + G1*X2 + a (error term) and then this Y2 = B2*X1 + G2*X2 + b (error term) these two error terms have to be uncorrelated with another error term, a. Don't know if that helps interpreting my question – Emil Krabbe Apr 16 at 7:46

So if we have sets $$M,N$$, then $$corr(M,N) = \{ corr(m,n) \mid m \in M, n \in N\},$$
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).