Answer to Question #267829 in Discrete Mathematics for aijsn

Question

2.     Let S = { :(a₁a₂a₃a₄∈ N and 0 ≤ ≤ 9 for each i = 1, 2, 3, 4}. In other words, S is the set of all 4-digit strings with each digit between 0 an 9. 

(a)  Show that the function f : S → S defined by f(a₁a₂a₃a₄) =a₄a₃a₂a₁  is a bijection.

(b)  (Note that the function reverses the string. For example, f(9527) = 7259) Find f^-1 .  Specifically, what is f^-1.(a₁a₂a₃a₄)?

Answer 1

(a)  Let us show that the function $f : S → S$ defined by $f(a_1a_2a_3a_4) =a_4a_3a_2a_1$   is a bijection.

Let $f(a_1a_2a_3a_4) =f(b_1b_2b_3b_4).$ Then $a_4a_3a_2a_1=b_4b_3b_2b_1,$ and hence $a_4=b_4, a_3=b_3,a_2=b_2,a_1=b_1.$ It follows that $a_1a_2a_3a_4=b_1b_2b_3b_4,$ and consequently, $f$ is injective. For any $b_1b_2b_3b_4\in S$ we have that $f(b_4b_3b_2b_1)=b_1b_2b_3b_4,$ and hence $f$ is surjective.

(b)  Let us show that $f^{-1}=f,$  that is $f^{-1}(a_1a_2a_3a_4)=a_4a_3a_2a_1.$ Indeed,

$f\circ f^{-1}(a_1a_2a_3a_4)=f(a_4a_3a_2a_1)=a_1a_2a_3a_4$ and

$f^{-1}\circ f(a_1a_2a_3a_4)=f^{-1}(a_4a_3a_2a_1)=a_1a_2a_3a_4.$