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Let $X_1,X_2$ be discrete random variables such that $H(X_1)<H(X_2)$ where $H()$ is the entropy. We know that for any random mapping $T$ which is invertible and independent of a random variable $X$ the following is true $$H(T(X))\geq H(T(X)|T)=H(T^{-1}T(X)|T)=H(X),$$ and we have $$H(T(X_1))\geq H(X_1)\text{ and }H(T(X_2))\geq H(X_2).$$ If $X_1$, $X_2$, $T(X_1 )$ and $T(X_2 )$ have the same support, is it true that $$H(T(X_1))\leq H(T(X_2))? $$

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Randomly mapping the symbols of $X$ into other symbols won't change the entropy of the variable (if the mapping is invertible). You are merely changing notation.

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