Let L be the set of all lines in a plane and $\mathrm{R}$ be the relation in $\mathrm{L}$ defined as $\mathrm{R}=\left\{\left(\mathrm{L}_{1}, \mathrm{L}_{2}\right): \mathrm{L}_{1}\right.$ is perpendicular to $\left. \mathrm{L} _{2}\right\}$. Show that $\mathrm{R}$ is symmetric but neither reflexive nor transitive.
Let L be the set of all lines in a plane and $\mathrm{R}$ be the relation in $\mathrm{L}$ defined as $\mathrm{R}=\left\{\left(\mathrm{L}_{1}, \mathrm{L}_{2}\right): \mathrm{L}_{1}\right.$ is perpendicular to $\left. \mathrm{L} _{2}\right\}$. Show that $\mathrm{R}$ is symmetric but neither reflexive nor transitive.
$R$ is not reflexive, as a line $L_{1}$ can not be perpendicular to itself, i.e., $\left(L_{1}, \,L_{1}\right)$ $\notin R$. $R$ is symmetric as $\left(L_{1}, L_{2}\right) \in R$
$\Rightarrow $ $L_{1}$ is perpendicular to $L_{2}$
$\Rightarrow $ $L_{2}$ is perpendicular to $L_{1}$
$\Rightarrow $ $\left(L_{2},\, L_{1}\right) \in R$
$R$ is not transitive. Indeed, if $L_{1}$ is perpendicular to $L_{2}$ and $L _{2}$ is perpendicular to $L _{3},$ then $L _{1}$ can never be perpendicular to $L _{3} .$ In fact, $L _{1}$ is parallel to $L _{3},$ ie., $\left( L _{1},\, L _{2}\right) \in R ,\left( L _{2}, L _{3}\right) \in R$ but $\left( L _{1}, L _{3}\right) \notin R$.
Similar Questions
The number of relations $R$ from an $m$-element set $A$ to an $n$-element set $B$ satisfying the condition$\left(a, b_1\right) \in R,\left(a, b_2\right) \in R \Rightarrow b_1=b_2$ for $a \in A, b_1, b_2 \in B$ is
The number of relations $R$ from an $m$-element set $A$ to an $n$-element set $B$ satisfying the condition$\left(a, b_1\right) \in R,\left(a, b_2\right) \in R \Rightarrow b_1=b_2$ for $a \in A, b_1, b_2 \in B$ is
- [KVPY 2009]
Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) R (u, v)$ if and only if $x v=y u .$ Show that $R$ is an equivalence relation.
Let $R$ be a relation on the set $A$ of ordered pairs of positive integers defined by $(x, y) R (u, v)$ if and only if $x v=y u .$ Show that $R$ is an equivalence relation.
Let $A =\{1,2,3,4, \ldots .10\}$ and $B =\{0,1,2,3,4\}$ The number of elements in the relation $R =\{( a , b )$ $\left.\in A \times A : 2( a - b )^2+3( a - b ) \in B \right\}$ is $.........$.
Let $A =\{1,2,3,4, \ldots .10\}$ and $B =\{0,1,2,3,4\}$ The number of elements in the relation $R =\{( a , b )$ $\left.\in A \times A : 2( a - b )^2+3( a - b ) \in B \right\}$ is $.........$.
- [JEE MAIN 2023]
The number of symmetric relations defined on the set $\{1,2,3,4\}$ which are not reflexive is
The number of symmetric relations defined on the set $\{1,2,3,4\}$ which are not reflexive is
- [JEE MAIN 2024]
Let $A$ be a set consisting of $10$ elements. The number of non-empty relations from $A$ to $A$ that are reflexive but not symmetric is
Let $A$ be a set consisting of $10$ elements. The number of non-empty relations from $A$ to $A$ that are reflexive but not symmetric is
- [KVPY 2020]