State with reason whether following functions have inverses:
(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)}
State with reason whether following functions have inverses:
(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)}
Function f:{1,2,3,4} → {10} defined as f={(1,10),(2,10),(3,10),(4,10)} From the given definition of f, we can see that f is a many to one function.
Function as f(1) =f(2) =f(3) =f(4) =10. Therefore, f is not one-one. Hence, function f does not have an inverse.
Function g:{5,6,7,8} → {1,2,3,4} defined as g ={(5,4),(6,3),(7,4),(8,2)} From the given definition of g it is seen that g is a many one.
Function as g(5) =g(7) =4, Therefore, g is not one -one.
Hence, function g does not have an inverse.
Function h:{2,3,4,5} → {7,9,11,13} defined as h={(2,7),(3,9),(4,11),(5,13)}
It is seen that all distinct elements of the set {2,3,4,5}have distinct images under h. Therefore, function h is one-one.
Also, h is onto, since for every element y of the set {7,9,11,13}, there exists an element x in the set {2,3,4,5} have distinct images under h. Thus, h is a one -one and onto function. Hence, h has an inverse.
Function f:{1,2,3,4} → {10} defined as f={(1,10),(2,10),(3,10),(4,10)} From the given definition of f, we can see that f is a many to one function.
Function as f(1) =f(2) =f(3) =f(4) =10. Therefore, f is not one-one. Hence, function f does not have an inverse.
Function g:{5,6,7,8} → {1,2,3,4} defined as g ={(5,4),(6,3),(7,4),(8,2)} From the given definition of g it is seen that g is a many one.
Function as g(5) =g(7) =4, Therefore, g is not one -one.
Hence, function g does not have an inverse.
Function h:{2,3,4,5} → {7,9,11,13} defined as h={(2,7),(3,9),(4,11),(5,13)}
It is seen that all distinct elements of the set {2,3,4,5}have distinct images under h. Therefore, function h is one-one.
Also, h is onto, since for every element y of the set {7,9,11,13}, there exists an element x in the set {2,3,4,5} have distinct images under h. Thus, h is a one -one and onto function. Hence, h has an inverse.
