Question #193283

For each of these relations on the set {1234}, decide

whether it is reflexive, whether it is symmetric, whether

it is antisymmetric, and whether it is transitive.

 {(22), (23), (24), (32), (33), (34)}


1
Expert's answer
2021-05-17T08:07:43-0400

We have given the set A= {1,2,3,41,2,3,4 }

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)R(2,4) \in R and (4,2)R(4,2) \notin R .

The relation R is not antisymmetric, because (2,3)R(2,3) \in R and (3,2)R(3,2) \in R , while 232 \ne 3 .

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

(2,2)Rand(2,3)R    (2,3)R(2,2)Rand(2,4)R    (2,4)R(2,3)Rand(3,2)R    (2,2)R(2,3)Rand(3,3)R    (2,3)R(2,3)Rand(3,4)R    (2,4)R(3,2)Rand(2,3)R    (3,3)R(3,2)Rand(2,4)R    (3,4)R(3,3)Rand(3,2)R    (3,2)R(3,3)Rand(3,3)R    (3,3)R(3,3)Rand(3,4)R    (3,4)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\\


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