Problem

Source: 2022 Taiwan TST Round 3 Mock P4

Tags: function, algebra, functional equation, domain, ming



Let X be the collection of all non-empty subsets (not necessarily finite) of the positive integer set N. Determine all functions f:XR+ satisfying the following properties: (i) For all S, TX with ST, there holds f(T)f(S). (ii) For all S, TX, there hold f(S)+f(T)f(S+T),f(S)f(T)=f(ST),where S+T={s+tsS,tT} and ST={stsS,tT}. Proposed by Li4, Untro368, and Ming Hsiao.