I'm not a mathematician and I'm trying to understand the properties of the binary entropy function. In particular, I would like to ask you, if the distributive law can be applied to the following expression: $(x_i - y_i) H(\frac{1}{2^{i-1}}) \quad s.t. \quad x_i H(\frac{1}{2^{i-1}}) - y_i H(\frac{1}{2^{i-1}})$, without any violation of the $H$'s properties? Thank you in advance!
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Yes; this is the distributive law. It's just $(x-y)h=xh-yh$. – Angina Seng Feb 02 '19 at 10:36
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Thank you for your prompt responses. So, if I understand well, the above expression does not violate any of the properties of the binary entropy function H and the distributive law is applied without any problem. – Yiota Feb 02 '19 at 10:43