I've come across a relation that satisfies all four of the relation properties: reflexive, symmetric, antisymmetric, and transitive.
First of all, is this even possible?? and if so, is it a very specific kind of relation?
The following is the relation:
Let A = {−7, −5, −3, −1, 1, 3, 5, 7}
R ={(a,b)|a^3 =b^3}
Any help is appreciated.
The equality relationship (also called the identity relationship) has all four properties.
reflexive: $\forall x \in X : x = x$
symmetric: $\forall a,b \in X : a = b \iff b = a$
antisymmetric: $\forall a, b \in X: (a=b) \wedge (b=a) \implies a = b$
transitive: $\forall a, b, c \in X: (a=b) \wedge (b=c) \implies a = c$