What is the size of the set of all possible functions?

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If I have a general function F(x) what is the size of the set containing all possible functions? Its obviously some infinity bigger than countable infinity but what size is it?

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The "set" of all possible functions is not a set, but a proper class. If you consider the set of possible functions from $A$ to $B$ then it depends on the cardinality of $A$ and $B$ via cardinal exponentiation.