We construct a symmetric invertible binary pairing function $F(m,n)$ on the set of positive integers with a property of $F(m,n)=F(n,m)$. Then we provide a complete proof of its symmetry and bijectivity, from which the construction of symmetric invert
ible binary pairing functions on any custom set of integers could be seen.
We decide completely the cycle structure of pure summing register (PSR) and complementary summing register (CSR). Based on the state diagram of CSR, we derive an algorithm to generate de Bruijn cycles from CSR inspired by Tuvi Etzions publication in
1984. We then point out the limitation in generalizations of extended representation we use in the algorithm proposed, with a proof of the fact that only PSR and CSR contain pure cycles all dividing n+1.
