$R = \{(2, 4), (4, 3), (2, 3), (4, 1)\}$
I know that $(2, 4) \in R$ and $(4, 3) \in R$ -> $(2,3)\in R$. But why my reference book said that the relation is not transitive?
And why this $R = \{(1, 1), (1, 2), (2, 2), (2, 1), (3,3)\}$ is transitive but the $(3,3)$ doesn't has any pair with it. Same like the R above that (4,1) doesn't has any pair with it. And also my lecturer said that we can use predicate calculus (math logic) according to know it is transitive or not. Is it true?
It is not transitive as $(2,4)$ and $(4,1)$ belongs to $R$ but $(2,1)$ does not belong to $R$.
Remember for transitivity you need $(a,b)\in R$ and $(b,c)\in R \implies (a,c) \in R$ for all $a,b,c \in R$.
Also the second relation is transitive as the above condition holds. For a relation to be transitive we do not need $(a,a) \in R$ for all $a$.