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Let X = {a, b, c, d} and give examples of a relation on X that is

a. reflexive and symmetrical, but not transitive,

b. reflexive and transitive, but not symmetrical,

c. transitive and symmetrical, but not reflective.

My Soultion for this is

a) (a,a),(a,a),(a,b),(b,a).

b)(a,a),(a,a) ,(a,b) ,(b,c)

c)(a,b),(b,a),(b,c)

But it is wrong what have i missed??

user3704516
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  • Reflexivity of relation $R$ on set $X$ asks for $\langle x,x\rangle\in R$ for every $x\in X$. You were told before here – drhab Nov 08 '16 at 14:42

1 Answers1

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Let $R$ be your relation on $X$. Also, $x\in A$ means that $x$ belongs to the set $A$.

First of all, you do not need to list $(a,a)$ twice as you are describing a set of tuples, i.e $\{(a,a),(a,a)\}=\{(a,a)\}$.

Second of all, in (b), if you want transitivity, then since $(a,b)\in R$ and $(b,c)\in R$ then $(a,c)$ need also be in $R$. This is the same kind of reason for why (c) is also wrong. Since you have $(a,b), (b,a)\in R$ then, by transitivity, you would have $(a,a)\in R$. Also, in (c), since $(a,b), (b,c)\in R$, again by transitivity, you would have $(a,c)\in R$.

Since this sounds like a typical homework problem, I will let you figure out if there is anything more wrong with it. Get back to me if you want, but I suggest that you recap the definitions of reflexive, symmetric and transitive relations.

Christopher.L
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