Problem 1 (2 pts): Give recursive definitions of the function f : {1, 2, ... } -> R, where (a) f(n) = 6n + 3, (b) f(n) = 2(n + 1), (c) f(n) = 3^n, (d) f(n) = n(n + 1)(n + 2).
Problem 2 (2 pts): How many one-to-one functions are there from a set of cardinality 10 to sets with the following cardinalities? (a) 7, (b) 11, (c) 15, (d) 20.
Problem 3 (3 pts): A palindrome is a string whose reversal is identical to the string. How many strings using lower-case letters a, b, ..., z of length n are palindromes?
Problem 4 (3 pts): Let n be a positive integer. Show that among any group of n + 1 (not necessarily consecutive) positive integers there are at least two with the same remainder when they are divided by n.
Extra Problem (4 pts.): Define recursively the function ONE defined on binary strings, that for a given string ω, ONE(ω) is equal to the number of occurrences of the character 1 in ω.
For instance
ONE(10010) = 2, ONE(00000) = 0, ONE(11111) = 5.
