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),(3, 2),(2, 1)}

Is R transitive? Symmetric? Reflexive?

Answer 1

Answer to Question #87692 – Math – Discrete Mathematics

Question

Let $A = \{1, 2, 3, 4\}$ and let $R$ be a relation on $A$ such that $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 3), (3, 2), (2, 1)\}$. Is $R$ Transitive? Symmetric? Reflexive?

Solution

A relation $R$ is **reflexive** if $(a, a) \in R$, for all $a \in A$.

$(1, 1), (2, 2), (3, 3), (4, 4) \in R$. Hence $R$ is reflexive.

A relation $R$ is **symmetric** if $(a, b) \in R$, then $(b, a) \in R$, for $a, b \in A$.

$(1, 2) \in R, (2, 1) \in R$.

$(2, 3) \in R, (3, 3) \in R$.

Thus $R$ is symmetric.

A relation $R$ is **transitive** if $(a, b) \in R$, $(b, c) \in R$, then $(a, c) \in R$, for all $a, b, c \in A$.

$(1, 2) \in R$ and $(2, 3) \in R$, but $(1, 3) \notin R$.

Thus, $R$ is not transitive.

**Answer**: it is not transitive, it is symmetric, it is reflexive.

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