Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $\mathrm{R}$ in the set $\mathrm{A}$ of human beings in a town at a particular time given by
$ \mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{x}$ and $ \mathrm{y}$ work at the same place $\}$
Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $\mathrm{R}$ in the set $\mathrm{A}$ of human beings in a town at a particular time given by
$ \mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{x}$ and $ \mathrm{y}$ work at the same place $\}$
$\mathrm{R} =\{( \mathrm{x} , \mathrm{y} ): \mathrm{x}$ and $\mathrm{y}$ work at the same place $\}$
$\Rightarrow(\mathrm{x}, \mathrm{x}) \in \mathrm{R}$ [as $\mathrm{x}$ and $\mathrm{x}$ work at the same place $]$
$\therefore \mathrm{R}$ is reflexive.
If $(\mathrm{x}, \mathrm{y}) \in \mathrm{R},$ then $\mathrm{x}$ and $\mathrm{y}$ work at the same place.
$\Rightarrow \mathrm{y}$ and $\mathrm{x}$ work at the same place.
$\Rightarrow(\mathrm{y}, \mathrm{x}) \in \mathrm{R}$
$\therefore \mathrm{R}$ is symmetric.
Now, let $(\mathrm{x}, \mathrm{y}),\,(\mathrm{y}, \mathrm{z}) \in \mathrm{R}$
$\Rightarrow \mathrm{x}$ and $\mathrm{y}$ work at the same place and $\mathrm{y}$ and $\mathrm{z}$ work at the same place.
$\Rightarrow \mathrm{x}$ and $\mathrm{z}$ work at the same place.
$\Rightarrow(\mathrm{x}, \mathrm{z}) \in \mathrm{R}$
$\therefore \mathrm{R}$ is transitive.
Hence. $\mathrm{R}$ is reflexive, symmetric, and transitive.
Similar Questions
Let $R\,= \{(x,y) : x,y \in N\, and\, x^2 -4xy +3y^2\, =0\}$, where $N$ is the set of all natural numbers. Then the relation $R$ is
Let $R\,= \{(x,y) : x,y \in N\, and\, x^2 -4xy +3y^2\, =0\}$, where $N$ is the set of all natural numbers. Then the relation $R$ is
- [JEE MAIN 2013]
In the set $A = \{1, 2, 3, 4, 5\}$, a relation $R$ is defined by $R = \{(x, y)| x, y$ $ \in A$ and $x < y\}$. Then $R$ is
In the set $A = \{1, 2, 3, 4, 5\}$, a relation $R$ is defined by $R = \{(x, y)| x, y$ $ \in A$ and $x < y\}$. Then $R$ is
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.
Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $\mathrm{R}$ in the set $\mathrm{Z}$ of all integers defined as $\mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{x}-\mathrm{y}$ is an integer $\}$
Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $\mathrm{R}$ in the set $\mathrm{Z}$ of all integers defined as $\mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{x}-\mathrm{y}$ is an integer $\}$
Let $R$ and $S$ be two equivalence relations on a set $A$. Then
Let $R$ and $S$ be two equivalence relations on a set $A$. Then