Assignment 5 Due Date: April 5, 2018. THURSDAY. Please submit your solutions thr
ID: 2291163 • Letter: A
Question
Assignment 5 Due Date: April 5, 2018. THURSDAY. Please submit your solutions through Canvas. Write your answers in word document and browse your file and upload it to canvas 1,fis the function from { a, b, c} to { 1, 2, 3 } such that/(a)-2, (b)-3,J(c)-1. Is finvertible, and if it is, what is its inverse? 2. Let fbe the function from fa, b, c, d; to {1, 2, 3} defined by f(a)- 3, f(b) 2, f(c) 1, and f(d) 3. Is fan onto function? 3. Let/be the function from x to y and X = { a, b, c } and Y-1, 2, 3) such that f(a)-3, f(b)-2, and f(c)-1. If f: X à Y is a function, then the inverse of fis the function f: Y à X show that function composition of both will give an Identity set (OR) In other words prove,??= 1 x and,fof'-1YExplanation / Answer
1. The function f is invertible because it is a one-to-one correspondance.
The inverse function f-1reverses the correspondance given by f, so,
f-1(1) = c
f-1(2) = a
f-1(3) = b
2. Yes, f is onto since all three elements of the co-domain are images of the elements in the domain. If the co-domain were changed to {1,2,3,4} , f would not be onto.
3. Given,
f(a) = 3
f(b) = 2
f(c) =1
The inverse function f-1reverses the correspondance given by f, so,
f-1(1) = c
f-1(2) = b
f-1(3) = a
(f-1o f)(a) = f-1(f(a)) = f-1(3) = a
(f-1o f)(b) = f-1(f(b)) = f-1(2) = b
(f-1o f)(c) = f-1(f(c)) = f-1(1) = c
so,
f-1o f = 1X
Now,
(f o f-1)(1) = f(f-1(1)) = f(c) = 1
(f o f-1)(2) = f(f-1(2)) = f(b) = 2
(f o f-1)(3) = f(f-1(3)) = f(a) = 3
so,
f o f-1 = 1Y