Answer:
(a) This function is neither one-to-one nor onto.
(b) This function is neither one-to-one nor onto.
(c) This relation is not a function.
(d) The function is onto but not one-to-one.
Step-by-step explanation:
Given information: A = {1, 2, 3, 4, 5} and B = {a, b, c, d}
A relation is a function if and only if there exist a unique output for each input.
One-to-one : A function is one-to-one if every element of the function's codomain is the image of at most one element of its domain.
Onto : A function is onto if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x) = y.
(a)
{(1, c) ,(2, c) ,(3, c) ,(4, c) ,(5, d)}
This relation is a function because all x-value has unique y-value.
The above function it not one-to-one because for more than one input we have same output (c have four domains).
The above function it not onto because all element of B are not have preimage (a and b have no preimage).
This function is neither one-to-one nor onto.
Similarly,
(b)
{(1, a ) ,(2, d ) ,(3, a ) ,(4, c ) }
This relation is a function because all x-value has unique y-value.
Here, a have more than one preimage and b have no preimage.
The function is neither one-to-one nor onto.
(c)
{(1, d ) ,(2, d ) ,(3, a ) ,(4, b ) ,(4, d ) ,(4, c )} .
For x=4 we have for than one value of y.
Therefore this relation is not a function.
(d)
{(1, c ) ,(2, b ) ,(3, a ) ,(4, d ) ,(5, a ) }
This relation is a function because all x-value has unique y-value.
Here, a have two preimage. So, this function is not one-to-one.
All elements of B have preimage. So, this function is onto.
The function is onto but not one-to-one.
