I'm revising for an exam - two questions:
i) What's the cardinality of the set of strict total orders on $R$?
This is a subset of the power set of $R$ (because $|R|=|R.R|$), so I know its cardinality is less than this - but I can't find a way to show its cardinality is equal to this, i.e. an injection from the power set into the set of strict total orders.
ii) What is the cardinality of the set of countable subsets of F, where F is the set of functions from $R$ to $R$?
Cardinality of F is same as cardinality of power set of $R$ (one way given by reasoning above, and the other way by the fact that $|P(R)|=|2^R|=|R^R|$. The set of countable subsets of this I am not sure how to calculate though.