MathLabs

Problem 4

Construct a function f:Q+→Q+f:\mathbb Q^+\to\mathbb Q^+ such that f(xf(y))=f(x)/yf(xf(y))=f(x)/y for all positive rational numbers x,yx,y.
Step 4 of 5: Define the function on paired primes
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f(pi)=qi,f(qi)=1pif(p_i)=q_i,\qquad f(q_i)=\frac1{p_i}
Detailed analysis

Partition the primes into two infinite sets S={p1,p2,…}S=\{p_1,p_2,\ldots\} and T={q1,q2,…}T=\{q_1,q_2,\ldots\}. Define f(pi)=qif(p_i)=q_i and f(qi)=1pif(q_i)=\frac1{p_i}, and extend multiplicatively to every positive rational using unique prime factorization.