Relations and Functions — Class 12 Maths Solution

(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.