How many distinct functions for a set containing four elements?

Question

How many distinct unary and binary functions can be defined on a set containing four elements?

Edit:

How many distinct unary and binary operations can be defined on a set containing four elements?

Answer 1

HINT: Let $A$ be a $4$-element set, say $A=\{a_1,a_2,a_3,a_4\}$. Imagine defining a unary function $f:A\to A$ one value at a time: you have $4$ choices for $f(a_1)$, $4$ choice for $f(a_2)$, $4$ choices for $f(a_3)$, and $4$ choices for $f(a_4)$. You can make those choices in $4\cdot4\cdot4\cdot4=4^4=256$ ways.

To define a binary function $f:A\times A\to A$, you’re still going to be making a bunch of $4$-way choices; how many such choices will you need to make, and what is the final number of binary functions on $A$?