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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 !!

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  • $\begingroup$ Hint: What does $\mathbb{N}^2$ look like? $\endgroup$ – John Douma Feb 25 '14 at 4:38
  • $\begingroup$ Is it each number related only with themselves ? $\endgroup$ – basratio Feb 25 '14 at 4:49
  • $\begingroup$ What does the graph of $\mathbb{N}^2$ look like? The answer below given by Fred Rickey lists the points in the relation. How would you graph those points? $\endgroup$ – John Douma Feb 25 '14 at 4:52
  • $\begingroup$ Each of them related to themselves ? but not between each other ? $\endgroup$ – basratio Feb 25 '14 at 4:53
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$x$ 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.

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  • $\begingroup$ Thanks ! In my head it was more $\alpha = \{ <5, 5>, <5, 4>, <5,3>, <5,2>, <5,1>, <5,0>\}$. So mixed up ! So this relation is reflexive, symmetric and antisymmetric, (not irreflexive, not asymmetric and not transitive..) ?? $\endgroup$ – basratio Feb 25 '14 at 4:46

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