I am sorry to tell you that your posting is meaningless.
What is the question in detail?
What are you asking?
The questions exactly is: Find a relatiom on the set [1,2,3,4] that is symmetric and antisymmetric, but is not the equality relation =.
I was trying draw a picture to show symmetry which would be a double headed arrow between all elements. The diagram for antisymetric means that it is only a one head arrow. I need to try to find a way to do show both relations.
Since S is nonempty, convince yourself that R is irreflexive. This should tell you that R cannot be an equivalence relation.
What now can be concluded about equality?
Convince yourself that R is vacuously both symmetric and antisymmetric (and incidently, transitive).