Q. 84.0( 4 Votes )

# 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.

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