An analysis of variance test (ANOVA) tests for a difference in means between two or more groups. In the case where there are more than two groups, the null hypothesis is written as the following:
$$ {H_0}: {\mu_1} = {\mu_2} ... = {\mu_k} $$
I have seen the alternative hypothesis for the ANOVA written something like this:
$$ {H_A}: {\mu_1} \neq {\mu_2} ... \neq {\mu_k} $$
with an inelegant written prose of "for any two means" or something similar.
The actual alternative hypothesis for the ANOVA test simply states that the comparison between any two means is unequal. Thus, the notation given above is not correct. What is the proper notation for the alternative hypothesis?