Given two sets $A$ and $B$, let $F(A, B)$ denote the set of all functions $f : A → B $ (no assumptions about injectivity or surjectivity – all functions from $A$ to $B$ are included). Let $|S|$ denote the cardinality of a set S.
a) If A and B are finite, with |A| = n and |B| = m, show that F(A,B) is finite and determine |F (A, B)|.
b) Let S be any finite set of cardinality greater than 1. Show that $F(N,S)$ is uncountable, but $F(S,N)$ is countable.
Best Answer
HINT: The product of finitely many countable sets is countable; and $F(\Bbb N,\{0,1\})$ is uncountable due to Cantor's theorem, finally show that if $|S|\leq|T|$ and $A$ is any set then $|F(A,S)|\leq|F(A,T)|$.