For independent $X,Y$, expand the <Entropic Ruzsa sum-difference inequality> from part (d):
$$
H(X+Y)-\frac12H(X)-\frac12H(Y)
\leq3\left(H(X-Y)-\frac12H(X)-\frac12H(Y)\right).
$$
Collecting the <information entropy> terms gives
$$
H(X+Y)+H(X)+H(Y)\leq3H(X-Y).
$$
Replacing $Y$ by $-Y$ interchanges sum and difference and preserves $H(Y)$, yielding the requested orientation
$$
H(X-Y)+H(X)+H(Y)\leq3H(X+Y).
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 224 / 1 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-224/1/e/solution