Twin Prime Conjecture (TP) 1st Way

My approach attempts to prove the Twin Prime Conjecture by constructing a large ambient set of all odd pairs that differ by 2, then arguing that because this ambient set is infinite, the subset consisting of twin prime pairs must also be infinite. I define: • A rule generating all odd pairs differing by 2: (3+2t,  5+2t),  t∈N • This produces the set K, which contains: o Twin prime pairs (e.g., (3,5), (5,7), (11,13), …) o Non twin-prime pairs (e.g., (7,9), (9,11), (25,27), …) I then partition K into: • TP: the subset of twin prime pairs • D: the subset of non twin-prime pairs Since K=TP∪D and K is infinite, I argue that both TP and D must be infinite. From this, you conclude that TP — the set of twin primes — is infinite.