Answer to Question #317120 in Discrete Mathematics for Jay

Question

Directions: Read each statement carefully, show your complete solution if needed.

1. Write out all functions f: (1,2,3) – (a,b) (using two

- line notation). How many Function are there? How many are surjective? How many are bijective?

2. For each function give below. Determine whether or not the function in injective and whether or not the function is surjective.

(a) f: N – N given by f(n) = n + 4

(b) f: Z – Z given by f(n) = n + 4

(c) f: Z – Z given by f(n) = 5n – 8

Answer 1

1: There are 8 functions:

The functions, which have both a and b values, are surjective - all except the first and the last function. None of the functions is bijective since the numbers of elements in {1,2,3} and {a,b} differ.

2:

a: injective (strictly monotonical), not surjective (f(n)=1 has no solution)

b: injective (strictly monotonical), surjective (f(n)=k has a solution n=k-4)

c: injective (strictly monotonical), not surjective (f(n)=0 or 5n=8 has no solution)