Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive.

(i) Relation R "is perpendicular to"

i.e., $R = \{ (x,\;y):x\;\;is\;\;perpendicular\;\;to\;y\}$

${l_1}$ is not perpendicular to ${l_1} \Rightarrow R$ is not reflexive

If ${l_1} \bot {l_2},$ then ${l_2} \bot {l_1} \Rightarrow$ R is symmetric

If ${l_1} \bot {l_2}\;\;and\;\;{l_2} \bot {l_3},$ then ${l_1}$ is not perndicular to ${l_3}.$

$\Rightarrow$ R is not transitive

Therefore we can say that R "is perpendicular to" is a symmetric but neither reflexive nor transitive.

(ii) Relation R $= \{(x, y) : x > y \}$

We know that$x > x$ is false. So, R is not reflexive.
If$x > y,$ then it does not imply that $y > x$. So, R is not symmetric.
If $x > y, y > z$ imply $x > z$. So, R is transitive.

Thus, R is transitive but neither reflexive nor symmetric.

(iii) Relation "is friend of "R $= \{(x, y) : x$ is a friend of y$\}$
x is a friend of x. Therefore, R is reflexive.

If x is a friend of y, then y is a friend of x. Therefore, R is symmetric.

If x is a friend ofy and y is a friend of z, then x cannot be friend of z.

Therefore, R is reflexive and symmetric but not transitive.

(iv) R is relation “is greater or equal to” i.e.,

$R = \{ (x,\;y):x \ge y\}$

$x \ge x\;\;is\;\;true.\;\;\;\;\therefore \;\;R\;\;is\;\;reflexive.$

If $x \ge y$ then it does not imply $y \ge x$ $\therefore \;\;R\;\;is\;\;not\;\;symmetric$

If $x \ge y$ then it does not imply $y \ge x$ Therefore, R is not symmetric

If $x \ge y,\;y \ge z \Rightarrow x \ge z$ Therefore, R is transitive.

Hence, R is reflexive and transitive but not symmetric.

(v) R is relation "is brother of " i.e.

R $= \{(x, y) : x$ is a brother of y$\}$

x is not a brother of x. So, R is not reflexive
If x is a brother ofy, then y is a brother ofx. So, R is symmetric

If x R y, and y Rz , i.e., x is brother ofy and y is brother of z

$\Rightarrow$ x is brother of z $\Rightarrow$ x R z $=$ R is transitive.

Hence, R is symmetric, transitive but not reflexive.