This is the general concept of math: to prove one set contains another set. In some cases you prove set equality by showing each side contains the other. A sim

This is the general concept of math: to prove one set contains another set. In some cases you prove set equality by showing each side contains the other. A simple example of this is: all even integers are divisible by two. That's two sets: the set of even numbers and the set of numbers divisible by two. For any x that is an even number, it is equal to 2k for some integer k. This implies that x/2 = 2k/2 = k. Since x/2 is an integer, it is divisible by 2 QED. It is also possible to prove that if an integer is divisible by 2, it is even. That's a different proof. The fact that you can solve problem A in terms of problem B doesn't always mean you can solve problem B in terms of problem A. Just because great minds think alike, doesn't mean people who think alike are great minds. All problems reducing to other problems in the technical sense are built on such foundations. The specific area you're thinking of is theoretical computer science. You might like the textbook "Introduction to the theory of computation" by sipser. It starts by showing the mutual-problem-reducibility of regular languages (AKA the language of deteinistic finite automata / regular expressions) and moves step by step into Turing machines which support general purpose programming languages. And beyond (languages which can only be conputed by hypothetical machines which we don't know how to build in reality.)