Give the mathematical expressions for the thermodynamic and statistical definitions of entropy.

In thermodynamics, entropy (S) is a measure of the disorder or randomness in a system. The thermodynamic definition of entropy is given by:

\[
dS = ____ _________ ________ ______ ________ _________ _____ ______.
_______ _______ _____ ______ ________ ___.
_______ ________ ______ ________ _____ ________ ______ _____ __________ ____.
______ ____ ______ _____ _________ __________ ___ _______ __________ _______.
___ ____ _______ _________ _____ ________ ________ _______ _______ ____ _________.
_________ _______ ____ ____ _______.
________ ________ _________ ________ _______.
______ _________ _________ _____ ______.
________ ______ ______ __________ _____ _________ __________ _______.
________ ____ _______ _______ __________ ___.
____ ________ ___ ___ ________ ____ ___ _______ ______ _________ __________ _____.
_________ ________ _________ _________ ________ ______ _____ _________ _____.
______ _____ _________ ________ _____ _____ _______ _____ _________ _____ _________.
____ __________ ____ ____ ______.
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