Answer to Question #193283 in Discrete Mathematics for Akmalzhon Nurmakha

Question

For each of these relations on the set {1, 2, 3, 4}, decide

whether it is reflexive, whether it is symmetric, whether

it is antisymmetric, and whether it is transitive.

 {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)}

Answer 1

We have given the set A= {$1,2,3,4$ }

The relation R is not reflexive, because R does not contain (1,1) and (4,4).

The relation R is not symmetric , because $(2,4) \in R$ and $(4,2) \notin R$ .

The relation R is not antisymmetric, because $(2,3) \in R$ and $(3,2) \in R$ , while $2 \ne 3$ .

The relation R is transitive, because if $(a,b) \in R$ and $(b,c) \in R$ then we also note that $(a,c) \in R$

$(2,2) \in R \hspace{2mm}and\hspace{2mm} (2,3) \in R \implies (2,3) \in R\\ (2,2) \in R \hspace{2mm}and\hspace{2mm} (2,4) \in R \implies (2,4) \in R\\ (2,3) \in R \hspace{2mm}and\hspace{2mm} (3,2) \in R \implies (2,2) \in R\\ (2,3) \in R\hspace{2mm} and \hspace{2mm}(3,3) \in R \implies (2,3) \in R\\ (2,3) \in R \hspace{2mm}and\hspace{2mm} (3,4) \in R \implies (2,4) \in R\\ (3,2) \in R \hspace{2mm}and \hspace{2mm}(2,3) \in R \implies (3,3) \in R\\ (3,2) \in R\hspace{2mm} and\hspace{2mm} (2,4) \in R \implies (3,4) \in R\\ (3,3) \in R \hspace{2mm}and\hspace{2mm} (3,2) \in R \implies (3,2) \in R\\ (3,3) \in R\hspace{2mm} and\hspace{2mm} (3,3) \in R \implies (3,3) \in R\\ (3,3) \in R\hspace{2mm} and\hspace{2mm} (3,4) \in R \implies (3,4) \in R\\$