$A = {1, 2, 3, 4}$
$R = { (1,2) (2,3) (3,4) }$
$R$ is not transitive, but if we add $(2,4)$ it will become transitive. my question is why do we ignore $(1,2)$ and $(2,3)$ since $(1,2)$ and $(2,3)$ satisfy $(a,b)$ and $(b,c)$.
Transitive relation in discrete mathematics
1
$\begingroup$
discrete-mathematics
equivalence-relations
2 Answers
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It won't become transitive even after adding $(2,4)$. We'll need to add $(1,3)$, as well as $(1,4)$ to make it transitive.
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R is transitive if for all $(a,b),(b,c) \in R$ we get that $(a,c) \in R$. Your claim that by adding $(2,4)$ that $R$ becomes transitive is false. Since $(1,2), (2,3) \in R$ we should have $(1, 3)$ for $R$ to be transitive. The smallest set containing $R$ that is transitive is in this case $\{(1,2), (2,3), (3,4), (2, 4), (1,3), (1, 4), \}$.