Question

Discuss the following relations for reflexivity, symmetricity and transitivity:

(i) The relation R defined on the set of all positive integers by “mRn if m divides n”.

(ii) Let P denote the set of all straight lines in a plane. The relation R defined by “lRm if l is perpendicular to m”.

(iii) Let A be the set consisting of all the members of a family. The relation R defined by “aRb if a is not a sister of b”.

(iv) Let A be the set consisting of all the female members of a family. The relation R defined by “aRb if a is not a sister of b”.

(v) On the set of natural numbers the relation R defined by “xRy if x + 2y = 1”.

Answer 1

(i) S = {set of all positive integers} 

(a) mRm ⇒ ‘m’ divides’m’ ⇒ reflexive 

(b) mRn ⇒ m divides n but nRm ⇒ n does not divide m (i.e.,) mRn ≠ nRm It is not symmetric 

(c) mRn ⇒ nRr as n divides r It is transitive

(ii) P = {set of all straight lines in a plane} 

lRm ⇒ l is perpendicular to m

(a) lRl ⇒ l is not perpendicular to l 

⇒ It is not reflexive 

(b) lRm ⇒ l is perpendicular to m 

mRl ⇒ m is perpendicular to l 

It is symmetric 

(c) l perpendicular to m ⇒ m perpendicular to n ⇒ l is parallel to n It is not transitive

(iii) A = {set of all members of the family} 

aRb is a is not a sister of b 

(a) aRa ⇒ a is not a sister of a It is reflexive 

(b) aRb ⇒ a is not a sister of b. 

bRa ⇒ b is not a sister of a. 

It is symmetric 

(c) aRb ⇒ a is not a sister of b. 

bRc ⇒ b is not a sister of c.

⇒ aRc ⇒ a can be a sister of c

It is not transitive.

(iv) A = {set of all female members of a family}

(a) aRa ⇒ a is a sister of a 

It is reflexive 

(b) aRb ⇒ a is a sister of b 

bRa ⇒ b is a sister of a 

⇒ It is symmetric 

(c) aRb ⇒ a is a sister of b bRc ⇒ b is a sister of c 

aRc ⇒ a can be sister of c It is not transitive.

(v) N= {1, 2, 3, 4, 5,…..} 

xRy if x + 2y = 1 R is an empty set 

(a) xRx ⇒ x + 2x = 1 ⇒ x = (1/3) ∉ N. It is not reflexive 

xRy = yRx ⇒ x + 2y = 1 

It does not imply that y + 2x = 1 as y = (1 - x)/2 It is not symmetric.

(b) -x = y ⇒ (-1, 1) ∉ N 

It is not transitive.