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Assessment: The following question was given to ChatGPT. Determine what's correct, fix all the mistakes, and assign a letter grade: A,B,C,D,F. Let us consider some new definitions. (a) Irreflexive

`AAxnot(xRx)`

(b) Antitransitivity:

`AAx,y,z(xRy^(^())yRz->not(xRz))`

Give a relation R on a set A such that it is neither reflexive, symmetric, transitive, antisymmetric, irreflexive nor antitransitive. To create a relation

`R`

on a set

`A`

such that it is neither reflexive, symmetric, transitive, antisymmetric, irreflexive, nor antitransitive, we can follow these steps: Reflexive:

`R`

is not reflexive if there exists at least one element

`ainA`

such that

`(a,a)!inR`

. Symmetric:

`R`

is not symmetric if there exist elements

`a,binA`

such that

`(a,b)inR`

but

`(b,a)!inR`

. Transitive:

`R`

is not transitive if there exist elements

`a,b,cinA`

such that

`(a,b)inR`

and

`(b,c)inR`

but

`(a,c)!inR`

. Antisymmetric:

`R`

is not antisymmetric if there exist elements

`a,binA`

such that

`(a,b)inR`

and

`(b,a)inR`

but

`a!=b`

. Irreflexive:

`R`

is not irreflexive if there exists at least one element

`ainA`

such that

`(a,a)inR`

.

`◻`

Antitransitive:

`R`

is not antitransitive if there exist elements

`a,b,cinA`

such that

`(a,b)inR`

and

`(b,c)inR`

and

`(a,c)inR`

. Consider the set

`A={1,2,3}`

. Define the relation

`R`

on

`A`

as follows:

`R={(1,2),(2,3),(3,1),(1,1)}`

Let's check each property: Reflexive:

`R`

is not reflexive because

`(2,2)!inR`

and

`(3,3)!inR`

. Symmetric:

`R`

is not symmetric because

`(1,2)inR`

but

`(2,1)!inR`

. Transitive:

`R`

is not transitive because

`(1,2)inR`

and

`(2,3)inR`

, but

`(1,3)!inR`

. Antisymmetric:

`R`

is not antisymmetric because

`(1,1)inR`

and

`(1,1)inR`

but

`1=1`

, so we need another pair to violate this, but for the given pairs, it doesn't. Irreflexive:

`R`

is not irreflexive because

`(1,1)inR`

. Antitransitive:

`R`

is not antitransitive because

`(1,2)inR,(2,3)inR`

, and

`(1,3)!inR`

, but we have

`(3,1)inR`

. Thus, the relation

`R={(1,2),(2,3),(3,1),(1,1)}`

on the set

`A={1,2,3}`

satisfies the condition of being neither reflexive, symmetric, transitive, antisymmetric, irreflexive, nor antitransitive.