Question

Given an example of a relation. Which is
A. Symmetric but neither reflexive nor transitive.
B. Transitive but neither reflexive nor symmetric.
C. Reflexive and symmetric but not transitive.
D. Reflexive and transitive but not symmetric.
E. Symmetric and transitive but not reflexive.

Answer 1

Hint: Let $A$ and $B$ be two sets. A relation from $A$ to $B$ is a subset of $A \times B$. Let $R$ be relation from $A$ into $B$, if $\left( {a,b} \right) \in R$, then we write it as $aRb$ and read it as $a$ is in relation to $b$.Reflexive relation is a relation $R$ in set $A$ if $aRb$ for all $a \in A$. Symmetric relation is a relation $R$ in set $A$ if $aRb \Rightarrow bRa$, that is, if $\left( {a,b} \right) \in R$ then $\left( {b,a} \right) \in R$. Transitive relation is a relation $R$ in set $A$ if $aRb$ and $bRc \Rightarrow aRc$, that is, if $\left( {a,b} \right)$ and $\left( {b,c} \right)$ belongs to $R$, then implies $\left( {a,c} \right)$ belongs to $R$.

Complete step-by-step answer:
A) The relation $R$ in the set $A = \left\{ {5,6,7} \right\}$ is defined by $\left\{ {\left( {5,6} \right),\left( {6,5} \right)} \right\}$ is symmetric, but neither reflexive nor transitive as $\left( {5,5} \right),\left( {6,6} \right),(7,7) \notin R$and $\left( {5,6} \right),\left( {6,5} \right) \in R$ but \[(5,5) \notin R\].

B) The relation $R$ in the set $A = \left\{ {1,2,3} \right\}$ defined by $R = \left\{ {\left( {1,3} \right),\left( {3,2} \right),\left( {1,2} \right)} \right\}$ is transitive, but neither reflexive nor symmetric.

C) The relation $R$ in the set $A = \left\{ {1,2,3} \right\}$ defined by $R = \left\{ {\left( {1,1} \right),\left( {1,2} \right),\left( {2,1} \right),\left( {2,3} \right),\left( {3,2} \right),\left( {2,2} \right),\left( {3,3} \right)} \right\}$ is reflexive and symmetric but not transitive.

D) The relation $R$ in the set $A = \left\{ {1,2,3} \right\}$ defined by $R = \left\{ {\left( {1,1} \right),\left( {1,2} \right),\left( {2,1} \right),\left( {2,2} \right)} \right\}$ is reflexive and transitive but not symmetric.

E) The relation $R$ in the set $A = \left\{ {1,2,3} \right\}$ defined by $R = \left\{ {\left( {1,1} \right),\left( {1,2} \right),\left( {2,1} \right),\left( {2,2} \right)} \right\}$ is symmetric and transitive but not reflexive.

Note: Let $L$ be the set of all straight lines in a plane. The relation $R$ in $L$ defined by ‘$x$’ is perpendicular to $y$ where ‘$x,y \in L$’ is symmetric but neither reflexive nor transitive. A relation $R$ in set $A$ is said to be an Equivalence relation is and only if $R$ is reflexive, $R$ is symmetric and $R$ is transitive.