An example of a relation that is symmetric and antisymmetric, but not reflexive.

Question

I am really stuck on if there is such an equation. The set given was A={1,2,3,4}. Is it even possible for a relation to be symmetric and antisymmetric, but not reflexive?

Answer 1

HINT: It doesn’t have to be a total relation. That is, its domain need not be all of $A$.

Answer 2

Antisymmetric: Assume a relation is antisymmetric unless, aRb and bRa and a≠b.

Symmetric: For all x and y ∈A,if xRy then yRx.

Reflexive: For all x∈A,xRx.

If A={1,2,3,4}, then {(1,1),(2,2)} is antisymmetric, symmetric and irreflexive.