# Give an example of a relation which is (ii) Transitive but neither reflexive nor symmetric.

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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 $\mathbf{transitive},$ if $(a_1,a_2) \in R$ and $(a_2,a_3) \in R \; \Rightarrow \;(a_1,a_3)\in R$ for all$\; a_1,a_2,a_3 \in A$
In a set of integers, we can observe that the Relation $R = \{(1,2), (2,4) (1,4)\}$ satisifes the properties of a transitive relation.
We can observe that while $(a,b) = (1,2) \in R$ and $(b,c) = (2,4) \in R$, which implies that $a=1, b=2, c=4$, we also see that $(a,c) = (1,4) \in R$. Hence $R$ is transitive.
Given that the Relation $R = \{(1,2), (2,4) (1,4)\}$, we can observe that if $a=1$, $(1,1) \not \in R$. Therefore, R is not reflexive.
Given that the Relation $R = \{(1,2), (2,4) (1,4)\}$, we can further observe that $(a,b) = (1,2) \in R$, but $(b,a) = (2,1)$ is not in $R$. Similarly $(4,2) \not \in R$ and $(4,1) \not \in R$.
Therefore, we can conclude that the Relation $R = \{(1,2), (2,4) (1,4)\}$ is transitive but not reflexive or symmetrical.

edited Mar 8, 2013