Just FYI, after reading Introducing Elixir and now 30% of the way through Programming Elixir 1.2, I can say that both books are great for beginners but I'm glad I read it in the order that I did since the examples and exercises in Programming Elixir 1.2 gradually get a little more challenging for a beginner. So, your foundation should be solid by the time you finish Introducing Elixir and come into Programming Elixir 1.2, which helps you get through the examples and exercises without feeling like you don't know jack. I haven't done the exercises, but I have read some of them and thought about how to solve them. I'll come back to actually doing them once I finish the book.
Both books dive into Elixir with very little Erlang, aside from how to use Erlang modules and maybe some cursory information regarding why Elixir does certain things the same or differently than Erlang. So, you don't need to have Erlang knowledge to understand Elixir. But, to do more advanced programming, it is probably a wise decision to read at least 1 or 2 books on Erlang. I'll probably dig into a few of the books below once I complete the books on Elixir that I have set aside:
Programming Erlang (heard great things)
Design for Scalability with Erlang/OTP (heard great things)
Learn You Some Erlang for Great Good (heard great things)
Erlang and OTP in Action (not heard much about this one)