Let $\mathrm{T}$ be the set of all triangles in a plane with $\mathrm{R}$ a relation in $\mathrm{T}$ given by $\mathrm{R} =\left\{\left( \mathrm{T} _{1}, \mathrm{T} _{2}\right): \mathrm{T} _{1}\right.$ is congruent to $\left. \mathrm{T} _{2}\right\}$ . Show that $\mathrm{R}$ is an equivalence relation.
Let $\mathrm{T}$ be the set of all triangles in a plane with $\mathrm{R}$ a relation in $\mathrm{T}$ given by $\mathrm{R} =\left\{\left( \mathrm{T} _{1}, \mathrm{T} _{2}\right): \mathrm{T} _{1}\right.$ is congruent to $\left. \mathrm{T} _{2}\right\}$ . Show that $\mathrm{R}$ is an equivalence relation.
$\mathrm{R}$ is reflexive, since every triangle is congruent to it self.
Further, $\left( \mathrm{T} _{1}, \,\mathrm{T}_{2}\right) \in \mathrm{R} \Rightarrow \mathrm{T} _{1}$ is congruent to $\mathrm{T} _{2} \Rightarrow \mathrm{T} _{2}$ is congruent to $\mathrm{T} _{1} \Rightarrow\left( \mathrm{T} _{2}, \mathrm{T} _{1}\right) \in \mathrm{R} .$
Hence, $\mathrm{R}$ is symmetric.
Moreover, $\left( \mathrm{T} _{1},\, \mathrm{T} _{2}\right),\left( \mathrm{T} _{2}, \,\mathrm{T} _{3}\right) \in \mathrm{R} \Rightarrow \mathrm{T} _{1}$ is congruent to $\mathrm{T} _{2}$ and $\mathrm{T} _{2}$ is congruent to $\mathrm{T} _{3} \Rightarrow \mathrm{T} _{1}$ is congruent to $\mathrm{T} _{3} \Rightarrow\left( \mathrm{T} _{1}, \,\mathrm{T} _{3}\right) \in \mathrm{R}$.
Therefore, $\mathrm{R}$ is an equivalence relation.
Similar Questions
If $R$ is an equivalence relation on a set $A$, then ${R^{ - 1}}$ is
If $R$ is an equivalence relation on a set $A$, then ${R^{ - 1}}$ is
Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $R$ in the set $A$ of human beings in a town at a particular time given by
$R =\{(x, y): x $ is father of $y\}$
Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $R$ in the set $A$ of human beings in a town at a particular time given by
$R =\{(x, y): x $ is father of $y\}$
Let $\mathrm{A}$ be the set of all students of a boys school. Show that the relation $\mathrm{R}$ in A given by $\mathrm{R} =\{(a, b): \mathrm{a} $ is sister of $\mathrm{b}\}$ is the empty relation and $\mathrm{R} ^{\prime}=\{(a, b)$ $:$ the difference between heights of $\mathrm{a}$ and $\mathrm{b}$ is less than $3\,\mathrm{meters}$ $\}$ is the universal relation.
Let $\mathrm{A}$ be the set of all students of a boys school. Show that the relation $\mathrm{R}$ in A given by $\mathrm{R} =\{(a, b): \mathrm{a} $ is sister of $\mathrm{b}\}$ is the empty relation and $\mathrm{R} ^{\prime}=\{(a, b)$ $:$ the difference between heights of $\mathrm{a}$ and $\mathrm{b}$ is less than $3\,\mathrm{meters}$ $\}$ is the universal relation.
Let $A =\{2,3,4\}$ and $B =\{8,9,12\}$. Then the number of elements in the relation $R=\left\{\left(\left(a_1, b_1\right),\left(a_2, b_2\right)\right) \in(A \times B, A \times B): a_1\right.$ divides $b_2$ and $a_2$ divides $\left.b_1\right\}$ is:
Let $A =\{2,3,4\}$ and $B =\{8,9,12\}$. Then the number of elements in the relation $R=\left\{\left(\left(a_1, b_1\right),\left(a_2, b_2\right)\right) \in(A \times B, A \times B): a_1\right.$ divides $b_2$ and $a_2$ divides $\left.b_1\right\}$ is:
- [JEE MAIN 2023]
Let $P = \{ (x,\,y)|{x^2} + {y^2} = 1,\,x,\,y \in R\} $. Then $P$ is
Let $P = \{ (x,\,y)|{x^2} + {y^2} = 1,\,x,\,y \in R\} $. Then $P$ is