Question

let S be the set of all points in a plane. R is a relation in S defined by R={(A,B):distance between A and B < 2 units}. show that R is reflexive and symmetric, but not transitive

Answer 1

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Answer 2

Given:

S be the set of all points in a plane.

R is a relation in S defined by,

R={(A,B):distance between A and B < 2 units}.

To show:

R is reflexive and symmetric, but not transitive.

Solution:

R = (A,B): distance between A and B < 2 units

We need to check this for reflexive, symmetric and transitive.

    1. Reflexive

• In this we compare A with A itself.
• If the condition remains satisfied, then R is reflexive.
• (A,A) = distance between A and A = 0 < 2 units.
• Hence condition is satisfied.
• Therefore, R is reflexive.

    2.  Symmetric

• In this we compare (A,B) and (B,A).

• If the condition remains satisfied, then R is reflexive.
• (A,B) = distance between A and B < 2
• (B,A) distance between B and A is same as distance between A and B.
• Hence R is symmetric.

     3. Transitive

• Here we compare A,B and another point C.
• We assume its true for (A, B) and (B,C).
• If it remains true for (A,B) ,then it is transitive.
• distance(A - B) < 2
• distance(B - C) < 2
• distance(A-C) < 4
• We can only say for sure that distance A and C is less than 4.
• Hence condition not satisfied.
• Therefore R is not transitive.

Hence showed that R is reflexive and symmetric, but not transitive.