Question

State with reason whether following functions have inverse (i) f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)} (ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)} (iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}

Answer 1

(i)

The function provided is f:{ 1,2,3,4 }→{ 10 } with f={ ( 1,10 ),( 2,10 ),( 3,10 ),( 4,10 ) } .

f( 1 )=10 f( 2 )=10 f( 3 )=10 f( 4 )=10

Since two or more values have the same image, the function is not one-one. Thus, the function does not have an inverse.

(ii)

The function provided is g:{ 5,6,7,8 }→{ 1,2,3,4 } with g={ ( 5,4 ),( 6,3 ),( 7,4 ),( 8,2 ) } .

g( 5 )=4 g( 6 )=3 g( 7 )=4 g( 8 )=2

Since two or more values have the same image, the function is not one-one.

Thus, the function does not have an inverse.

(iii)

The function provided is h:{ 2,3,4,5 }→{ 7,9,11,13 } with h={ ( 2,7 ),( 3,9 ),( 4,11 ),( 5,13 ) } .

The equation to find that the function is one-one or not is,

h( 2 )=7 h( 3 )=9 h( 4 )=11 h( 5 )=13

Since all the values have different images, the function is one-one.

The function h( x ) is onto for every element y of the set { 7,9,11,13 } , there exists an element x in the set { 2,3,4,5 } such that h( x )=y .

As the function h( x ) is one-one function and onto,therefore, the function has an inverse.