Question

Let A = {1, 2, 3, 4} and let R be a relation on A such that R = {(1, 1),(2, 2),(3, 3),(4, 4),(1, 2),(2, 3),(1, 3)}

Is R transitive? Symmetric? Reflexive?

Answer 1

Answer to Question #87694 – Math – Discrete Mathematics

Question

Let $A = \{1, 2, 3, 4\}$ and let $R$ be a relation on $A$ such that $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 3), (1, 3)\}$

Is R transitive? Symmetric? Reflexive?

Solution

For a relation $R \subset A \times A$ we have

(i) $R$ is reflexive if for each $a \in A$ we have $(a, a) \in R$,

(ii) $R$ is symmetric if for each $(a,b)\in R$ we have $(b,a)\in R$,

(iii) $R$ is transitive if $(a,b)\in R$ and $(b,c)\in R$ implies $(a,c)\in R$.

Thus, for the set $A = \{1,2,3,4\}$ and relation $R = \{(1,1),(2,2),(3,3),(4,4),(1,2),(2,3),(1,3)\}$ we have

(i) $R$ is reflexive since $(1,1),(2,2),(3,3),(4,4)\in R$,

(ii) $R$ is not symmetric since $(1,2)\in R$ but $(2,1)\notin R$,

(iii) $R$ is transitive since there is no $(a,b)\in R$ and $(b,c)\in R$ such that $(a,c)\notin R$.

Answer: transitive, not symmetric, reflexive.

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