Answer to Question #129379 in Discrete Mathematics for jaya

Question

(a) Let A = {a,b,c,d,e,f} and B = {1,2,3,4,5,6}. Determine whether each of following functions from A to B are invertible. If it is an invertible, find it’s inverse.
(i) f1 = {(a,1), (b,2), (c,3), (d,4), (e,5), (f,6)}
(ii) f2 = {(a,2), (b,5), (d,2), (c,3), (e,5), (f,6)}
(iii) f3 = {(b,3), (d,6), (a,1), (c,3), (e,4), (f,5)}
(iv) f4 = {(a,6), (b,5), (c,1), (f,2), (d,4), (e,3)}

Answer 1

Given A and B are two set with same number of elements, then if function A to B is one-one then that function is onto and hence invertible.

Now, f1 = {(a,1), (b,2), (c,3), (d,4), (e,5), (f,6)} and f4 = {(a,6), (b,5), (c,1), (f,2), (d,4), (e,3)} are invertible function, since they are one-one functions.

Given f2 are not one-one function since a and d have same image 2 under f2

and f3 are not one-one function since b and c have same image 3 under f3.

Inverse of function f1 = {(1,a), (2,b), (3,c), (4,d), (5,e), (6,f)}.

Inverse of function f2 = {(6,a), (5,b), (1,c), (2,f), (4,d), (3,e)}.