What Are the Properties of Binary Relations in Sets?

[thomas49th]

Hi, I'm struggling about with binary relations in sets. Can somebody check over and answer my questions about these sets:

Given set A = {1,2,3}

Provide one example each of a relation with the following properties where the cardinality of the relationship should be at least one in all cases:

(i) {(1,1),(2,2),(3,3),(2,3)} this is reflexive because all elements of A act upon itself it's no symmetrical as there exists no (3,2). I don't know whether it's transitive though as (2,3) is made from (2,2) and (3,3) so doesn't that make it transitive?
I ask whether the cardinality of (i) is 4 or 8?

(ii) Transitive but not reflexive or symmetric:
{(1,1),(2,2),(1,2)} it's not reflexive as not all elements for A act upon themselves, it's not symmetric as there is no (2,1). It's transitive though because (1,1) and (2,2) form (1,2).

(iii) Symmetric but not transitive or reflexive:
{(1,1),(2,2),(1,2),(2,1)}
Symmetric as (1,2) and (2,1) not reflexive as no (3,3) BUT what about transitivity? doesn't (1,1) and (2,2) make (1,2)?

I will post more up later once these are sorted.

Thanks
Thomas

 

[lippyka]

For (i), the cardinality is 4. For (ii), it is transitive as (1,1) and (2,2) make (1,2). For (iii), it is not transitive because (1,1) and (2,2) do not make (1,2).