Let S be the set of all points in a plane and let R be a relation in S defined by R = {(A, B) : d(A, B) < 2 units}, where d(A, B) is the distance between the points A and B.
Show that R is reflexive and symmetric but not transitive.
Given that, ∀ A, B ∈ S, R = {(A, B) : d(A, B) < 2 units}.
Now,
R is Reflexive if (A,A) ∈ R ∀ A ∈ S
For any A ∈ S, we have
d(A,A) = 0, which is less than 2 units
⇒ (A,A) ∈ R
Thus, R is reflexive.
R is Symmetric if (A, B) ∈ R ⇒ (B,A) ∈ R ∀ A,B ∈ S
(A, B) ∈ R
⇒ d(A, B) < 2 units
⇒ d(B, A) < 2 units
⇒ (B,A) ∈ R
Thus, R is symmetric .
R is Transitive if (A, B) ∈ R and (B,C) ∈ R ⇒ (A,C) ∈ R ∀ A,B,C ∈ S
Consider points A(0,0),B(1.5,0) and C(3.2,0).
d(A,B)=1.5 units < 2 units and d(B,C)=1.7 units < 2 units
d(A,C)= 3.2 ≮ 2
⇒ (A, B) ∈ R and (B,C) ∈ R ⇒ (A,C) ∉ R
Thus, R is not transitive.
Thus, R is reflexive, symmetric but not transitive.