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
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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.
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