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# In a set of integers, is the Relation $R = \{(1,2), (2,1)\}$ symmetrical, transitive or reflexive?

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Toolbox:
• A relation R in a set A is called $\mathbf{ reflexive},$ if $(a,a) \in R\;$ for every $\; a\in\;A$
• A relation R in a set A is called $\mathbf{symmetric}$, if $(a_1,a_2) \in R\;\Rightarrow\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$
• A relation R in a set A is called $\mathbf{transitive},$ if $(a_1,a_2) \in R$ and $(a_2,a_3) \in R \; \Rightarrow \;(a_1,a_3)\in R$ for all$\; a_1,a_2,a_3 \in A$
A relation R in a set A is called $\mathbf{symmetric}$, if $(a_1,a_2) \in R\;\Rightarrow\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$
Given R, We can observe that the Relation $R = \{(1,2), (2,1)\}$ satisifes the properties of a symmetrical relation.
Here, $(1,2) \in R \rightarrow (2,1) \in R$. Hence it is symmetric.
Given $R = \{(1,2), (2,1)\}$, we can observe that if $a=1$, $(1,1) \not \in R$. Therefore, R is not reflexive.
Given $R = \{(1,2), (2,1)\}$ we can further observe that while $(a,b) = (1,2) \in R$ and $(b,c) = (2,1) \in R$, which implies that $a=1, b=2, c=1$, we see that $(a,c) = (1,1) \not \in R$. Hence $R$ is not transitive.
Therefore, given a set of integers for instance, the Relation $R = \{(1,2), (2,1)\}$ is the example of a relation that is symmetric, but not reflexive or transitive.
answered Mar 8, 2013