Solution to 2. Let S = { :(a1a2a3a4∈ N and 0 ≤ ≤ 9 for each i … - Sikademy
Author Image

Archangel Macsika

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)?

The Answer to the Question
is below this banner.

Can't find a solution anywhere?


Get the Answers Now!

You will get a detailed answer to your question or assignment in the shortest time possible.

Here's the Solution to this Question

(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.

Related Answers

Was this answer helpful?

Join our Community to stay in the know

Get updates for similar and other helpful Answers

Question ID: mtid-5-stid-8-sqid-830-qpid-715