Question

SUPPOSE A = {a,b,c,d,e,f},and R = {(a,f),(b,e),(c,d),(d,a),(e,b),(f,e)} 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

Answer 1

(I) Yes

(II) Yes

(III) No

(IV) No

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:
`\forall y \in B : \exists x \in A / f(x)=y`.

In this case,
`\\exists x \in A / xRc`. 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:
`x \\eq y \Rightarrow f(x) \\eq f(y)`.

In this case, both
`b` and
`f` relate to
`e`, so the function is not one to one.