1. ## Re: Reflexive And Symmetric

Originally Posted by Hartlw
The possible relations on {1.2.3} are:
{(1,1), (2,2), (3,3)}
{(1,2), (2,3), (1,3)}
{(2,1) (3,2), (3,1)}
{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)}
{(1,1), (2,2), (3,3), (2,1), (3,2), (3,1)}
You missed a whole lot of the relations on $\{1,2,3\}$.
There are $2^9=512$ possible relations.
Any subset of $\{1,2,3\}\times\{1,2,3\}$.
As I said you don't know what a relation is.

2. ## Re: Reflexive And Symmetric

Originally Posted by Plato
You missed a whole lot of the relations on $\{1,2,3\}$.
There are $2^9=512$ possible relations.
Any subset of $\{1,2,3\}\times\{1,2,3\}$.
As I said you don't know what a relation is.
Not if 1,2,3 are integers, or you accept meaningless definitions which are really not relations. Frankly, I wavered on acceptable subsets. I don't believe they apply for the integers, depending on the definition of "relation," for example {(1,1), (1,2)}, but that's a topic for another discussion. In any event, your post 2 set is not a possible relation for the integers.

3. ## Re: Reflexive And Symmetric

Originally Posted by Hartlw
Not if 1,2,3 are integers, or you accept meaningless definitions which are really not relations. Frankly, I wavered on acceptable subsets. I don't believe they apply for the integers, depending on the definition of "relation," for example {(1,1), (1,2)}, but that's a topic for another discussion. In any event, your post 2 set is not a possible relation for the integers.
If $\mathcal{R}=\{(1,1),~(2,2),~(3,3),~(1,2),~(2,1)\} \subseteq (\mathbb{Z}\times\mathbb{Z})$ then it is a relation in the integers.

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