Browse Questions

# Determine whether Relation $R$ is reflexive, symmetric and transitive: Relation $R$ in the set $A$ of all human beings in a town at a particular time given by $R=\{(x,y):\; x$ is exactly $7\;cm$ taller than $y\}$

Toolbox:
• A relation R in a set A is called reflexive. if $(a,a) \in R\;$ for every$\;a \in A$
• A relation R in a set A is called symmetric. if $(a_1,a_2) \in R\;=>\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$
• A relation R in a set A is called transitive. if $(a_1,a_2) \in\;and\; R(a_2,a_3)=>that \;(a_1,a_3)\in R for\; all\; a_1,a_2,a_3 \in A$
Given set $A$ of all human beings in a town at a particular time and $R=\{(x,y):\; \text{x is exactly 7cm taller than y}\}$
If $x$ is exactly 7cm taller than $y$, $\rightarrow$ obviously $(x,x) \not \in R$, as $x$ cannot be 7cm taller han $x$.Hence $R$ is not reflexive.
If $x$ is exactly 7cm taller than $y$, $y$ obviously cannot be exactly 7cm taller than $x\; \rightarrow (x,y) \in R$, but $(y,x) \not \in R$. Therefore $R$ is not symmetric.
If $x$ is exactly 7cm taller than $y$, and $y$ is exactly 7cm taller than $z \; \rightarrow$ $x$ is obviously exactly 7+7 = 14cm taller than $z$ and not exactly 7cm taller than $z$.
Therefore, $(x,y) \in R, (y,z) \in R$ but $(x,z) \not \in R$. Hence $R$ is not transitive.