For purposes of this exercise, we will define random as having all elements equally likely to be chosen. So a function that selects a 100-bit number will be random if every 100-bit number is equally likely to be chosen. Using this definition, if we look at the function Ă˘â‚¬Ĺ“+Ă˘â‚¬Âť and we have two inputs, x and y, then the output will be random if at least one of x and y are random. For instance, y can always be 51, and yet the output will be random if x is random. For the following functions, find sufficient conditions for x, y, and z under which the output will be random:
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