# Trying to determine if this relation is reflexive, symmetric, antisymmetric and transitive

Let A be the set of all people who have ever lived. For x, y ∈ A, xRy if and only if x and y were born at least 30 days apart

I want to determine whether the relation xRy is reflexive, transitive, antisymmetric, and/or symmetric so I can go on to determine whether it is a partial order an equivalence relation, or both by means of simple counterexamples or short proofs.

• So far the conclusion I have come to is that the relation xRx is not reflexive as x can not have been born 30 days apart from x

• I also believe the relation xRy is symmetric as if x was born 30 days apart from y(xRy) then it follows that y was born 30 days apart from x (yRx)

I understand what it means for a relation to be an equivalence relation (symmetric, transitive, and reflexive) and what it means for a relation to be a partial order (antisymmetric, transitive, and reflexive) but I am unsure how to show these properties in this question.

I am unsure if the deductions I have made are correct and I am also unsure how to show if the relation is transitive and/or antisymmetric. Any help would be highly appreciated!

Your deductions about the reflexive and symmetric parts are correct. Note that the relation is not transitive. Assume $$x$$ and $$y$$ in $$1000-1100$$, and $$z$$ lived in $$1130-1230$$. Then $$xRz$$ and $$zRy$$, but $$x$$ is not related to $$y$$!
Note also that this relation is not antisymmetric. With the same example, $$xRz$$ and $$zRx$$, but $$x\neq z$$.