Is R an Identity Relation on A?

[oriel1]

Let A= {1,2,3}.
Let R= {<1,1>,<2,2>}.

I(A) (Identity Realtion) on A >(def)> {<x,x>|x \(\displaystyle \in\) A}
So that mean : \(\displaystyle \forall\) <x,x> x \(\displaystyle \in\) A
(That how I understood it)

My question:
Is R is identity relation on A ?

Thank you !

 

[Deveno]

No. Think about (3,3).

 

[oriel1]

No. Think about (3,3).

Ok Actually R={<1,1>,<2,2>,<3,3>} is identity relation on A for sure.
But what prevent from R= {<1,1>,<2,2>} to bo identity on A?
It not writed \(\displaystyle \forall\) x \(\displaystyle \in\) A.

 

[Deveno]

The definition (yours, not mine) says:

$I(A) = \{(x,x)\mid x \in A\}$.

However, $3 \in A = \{1,2,3\}$, but $(3,3) \not\in R$.

That is, $(3,3)$ is a pair with $3 \in A$, and thus $(3,3)$ fulfills the requirements to be an element of $I(A)$. Most texts define the identity relation as the smallest possible equivalence relation on a given set, and your relation fails the reflexive test.

 

[oriel1]

Thank you. now i understand it.