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It is needed to prove that inequality

$I(a:b) \le I(a:b|x) + I(a:b|y) + I(x:y)$ (where I(a:b) is mutual information and I(a:b|x) is conditional mutual information)

is true not for all sets of random variables a,b,x,y.

I tried to do it by using formulas for mutual information and conditional mutual information and by expressing all in terms of entropy and using similar approval for enthropies:

"The next inequality is true not for all sets of random variables a,b,c

$2 \cdot H(a,b,c) \le H(a,b) + H(a,c|b) + H(b,c|a)$"

But I did not manage. Please help to understand how to prove it. Thank you in advance.

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