1. ## Relations question

My textbook does an erratic job of presenting relations to the reader, my professor is of no help, and I'm thoroughly puzzled, particularly with this question.

For each of the following relations defined on the set {1,2,3,4,5}, determine whether the relation is reflexive, irreflexive, symmetric, antisymmetric, and/or transitive:

a. R = {(1,1),(2,2),(3,3),(4,4),(5,5)}

b. R = {(1,2),(2,3),(3,4),(4,5)}

c. R = {(1,1),(1,2),(1,3),(1,4),(1,5)}

d. R = {(1,1),(1,2),(2,1),(3,4),(4,3)}

e. R = {1,2,3,4,5} x {1,2,3,4,5}

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Could someone walk me through some of these so I can understand the logic? I understand the essential meaning of a relation and how to apply these terms to the classical symbolic relations, but this is puzzling to me.

Regards.

2. Hi
A relation $\mathcal{R}$ can be completely determined by its graph, which here is named $R,$ and which is a set of ordered pairs such that: $(a,b)\in R\Leftrightarrow a\mathcal{R}b.$ (I guess that most people will say that a relation and it's graph are the same thing)

For 1), for instance, a way to check wether R is a relation or not:

Reflexivity: $\forall a\in\{1,2,3,4,5\},\ (a,a)\in R$ . (i.e. aRa). That's true, thus R is reflexive.

Irreflexivity: $\forall a\in\{1,2,3,4,5\},\ (a,a)\notin R$ . (i.e. no(aRa) ) It's wrong, for example (1,1) is in R.

symmetry: $\forall a,b\in\{1,2,3,4,5\},\ (a,b)\in R\Rightarrow (b,a)\in R$ . That's the case.

antisymmetry: $\forall a,b\in\{1,2,3,4,5\},\ ((a,b)\in R\wedge a\neq b)\Rightarrow\neg((b,a)\in R)$ . Again, that's true (since there is no (a,b) in R with a $\neq$b)

transitivity: $\forall a,b,c\in\{1,2,3,4,5\},\ ((a,b)\in R\wedge (b,c)\in R)\Rightarrow (a,c)\in R$ . True, for the same reason given for antisymmetry.

(That relation is particular, it's the equality)