Suppose a = {a, b,c, d,e, f},and r = {(ae)} state with reasons whether the relation is a function from a to a,(ii) an everywhere defined function (iii) an onto function (iv) a one to one function

(I) Yes

(II) Yes

(III) No

(IV) No

Step-by-step explanation:

(I) The relation is a function from A to A as it satisfies both existence and uniqueness: every element of A (domain) is related to only one element of A (codomain).

(II) The relation is an everywhere defined function as every element of A is related to an element of A.

(III) The relation is not an onto (surjective) function.

Let's recall the definition of an onto function:

An onto function f : A -> B satisfies: .

In this case, . So, the function is not onto.

(IV) The relation is not a one to one (injective) function.

Let's recall the definition of a one to one function:

A one to one function f : A -> B satisfies: .

In this case, both and relate to , so the function is not one to one.

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step-by-step explanation: