Let a = {1, 2, 3, 4, 5} and b = {a, b, c, d}. for each of the following relations fromn a to b, answer these questions: is it a function from a to b? if it is a function from a to b, is it one-to-one? is it onto? justify.
(a) {(1, c ) ,(2, c ) ,(3, c ) ,(4, c ) ,(5, d ) }
(b) {(1, a ) ,(2, d ) ,(3, a ) ,(4, c ) }
(c) {(1, d ) ,(2, d ) ,(3, a ) ,(4, b ) ,(4, d ) ,(4, c ) } .
(d) {(1, c ) ,(2, b ) ,(3, a ) ,(4, d ) ,(5, a ) }

(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.

answer: the answer is one of

step-by-step explanation:   we are given to write a proper fraction where the whole is divided into an even number of equal parts less than ten, and an odd number of those parts exist.

an even number less than 10 may be one of 2, 4, 6 or 8. so, the whole may be one of these four evens.

now, an odd number of these parts must exist, so the numerator must be an odd number, one of 1, 3, 5 or 7. it cannot be 9 because then the fraction will become improper.

thus, our fraction may be one of the following

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