Understanding relations when it's about $\langle{x,x}\rangle$ instead of $\langle{x,y}\rangle$

Question

I have difficult to understand relations when we talk about $\langle{x,x}\rangle$ instead of $\langle{x,y}\rangle$ .. it's hard for me to realize for example is the following relation is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive..

$\alpha = \{\langle{x,x}\rangle \in \mathbb{N}^2 \mid x \leq 5\} $

I cannot figure out how it would be the graph for this (the kind of graph we use to show relations, the big circle representing the set and the elements inside).. any tip to make it clearer in my head ? Thanks !!

Answer 1

$x$ can only assume the values 5, 4, 3, 2, 1, and 0 (if you consider 0 to be a natural number). Thus $\def\p#1{\langle {#1},{#1}\rangle}\alpha = \{ \p5, \p4, \p3, \p2, \p1, \p0\}$. This is an equivalence relation.