I'm studying properties of relations and there is one area that i'm kind of unsure about regarding the properties of asymmetry and anti-symmetry.

Suppose R = {(1,2),(3,4),(2,2)}

It would follow that R is:

Not reflexive, Not irreflexive, Not symmetric,

I would say it is not asymmetric and not antisymmetric also, but I get hung up on the (2,2) element.

Does (2,2), or any reflexive ordered pair, count as a = 2 and b = 2, such that a R b and b R a ?

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    $\begingroup$ It's transitive and antisymmetric. $\endgroup$ – Git Gud Feb 7 '15 at 17:02
  • $\begingroup$ So what you are saying is (2,2) means (a,a) and it does not mean (a,b) ? $\endgroup$ – Mike Feb 7 '15 at 17:05

Not reflexive Correct. Not every element is related to itself.

Not irreflexive Correct. Since we have $(2, 2)$. Irreflexivity requires that no element should be related to itself.

Not symmetric Correct. We have $(3, 4)$ but not $(4, 3)$.

Antisymmetric Correct, the only instance where $(a, b) \in R$ and $(b, a) \in R$ is $a = b$ (the pair$(2, 2))$

Not transitive ?

It is transitive because $(1, 2)$ and $(2, 2)$.

Not that this relation is defined on a set, which has a unique element 2, it seems to be the point of confusion from the comments.

  • $\begingroup$ hey axiom, thanks for the reply. Yes I think that is my point of confusion. Suppose that the relation is defined from set A = {1,2,3,4,5}. would (2,2) mean (a,b) or (a,a). $\endgroup$ – Mike Feb 7 '15 at 17:17
  • $\begingroup$ @Mike it would mean (a, a). $\endgroup$ – axiom Feb 7 '15 at 17:19
  • $\begingroup$ This would mean that the relation is Not antisymmetric then because there is no a = b it is a = a ? $\endgroup$ – Mike Feb 7 '15 at 17:22
  • $\begingroup$ @Mike The moment I add a pair $(a, b)$ and $(b, a)$, the relation ceases to be antisymmetric. Make sure you pay attention to the "for all" in the definition of antisymmetric. $\endgroup$ – axiom Feb 7 '15 at 17:24
  • $\begingroup$ Boom, that comment just made it clear for me haha. Thanks! $\endgroup$ – Mike Feb 7 '15 at 17:28

Taken any $a$, $b$ and $c$, we have:

$(a,b)$ and $(b,a)$ $\in R$ $\Rightarrow$ $a = b = 2$ because there are no other such couples in $R$. This shows $R$ to be anti-symmetric.

$(a,b)$ and $(b,c)$ $\in R$ $\Rightarrow$ $a = 1$, $b=c=2$ or $a=b=c=2$. In both cases: $(a,c) \in$ $R$. This is why $R$ is transitive.


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