Problem $4.$ Activity Independence.

Consider a research team studying the activity patterns of individuals within a social network. They maintain a matrix $\backslash ($ A $\backslash ),$ where the rows represent different individuals and the columns represent various activities. Each entry $\backslash ($ A$[$i$,$ j$]\backslash )$ captures the frequency with which individual $\backslash ($ i $\backslash )$ has engaged in activity $\backslash ($ j $\backslash ).$ Here's an example of such a matrix:

One objective of the researchers might be to identify a group of individuals with non$-$overlapping activity profiles. Specifically, a subset $\backslash ($ S $\backslash )$ of individuals is considered activity$-$distinct if no two members of $\backslash ($ S $\backslash )$ have participated in the same activity $($i$.$e$.,$ for any activity, at most one person in $\backslash ($ S $\backslash )$ has engaged in it$).$ Such an activity$-$distinct group might serve as a representative sample for further study.

This leads to the Activity$-$Distinct$-$Subset Problem: Given an $\backslash ($ m $\backslash$times n $\backslash )$ matrix $\backslash ($ A $\backslash )$ as described above and a number $\backslash ($ k $\backslash$leq m $\backslash ),$ is it possible to find an activity$-$distinct subset of at least $\backslash ($ k $\backslash )$ individuals?

Prove Activity$-$Distinct$-$Subset is NP$-$complete.