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Home » 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.
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.
CBSE | Class 12 | Exercise 1B | Maths | Relations | RS Aggarwal
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.

Solution:

\mathrm{R}=\{(\mathrm{A}, \mathrm{B}): \mathrm{d}(\mathrm{A}, \mathrm{B})<2 units \}, where \mathrm{d}(\mathrm{A}, \mathrm{B}) is the distance between the points \mathrm{A} and \mathrm{B}. (As given)

Reflexivity:
Suppose \mathrm{A} be an arbitrary element of \mathrm{S}
\begin{array}{l} \Rightarrow \mathrm{d}(\mathrm{A}, \mathrm{A})<2 \\ \Rightarrow(\mathrm{A}, \mathrm{A}) \in \mathrm{R} \end{array}
Therefore, R is reflexive.

Symmetric:
Suppose A and B \in S, such that (A, B) \in R
\begin{array}{l} \Rightarrow \mathrm{d}(\mathrm{A}, \mathrm{B})<2 \\ \Rightarrow \mathrm{d}(\mathrm{B}, \mathrm{A})<2 \\ \Rightarrow(\mathrm{B}, \mathrm{A}) \in \mathrm{R} \end{array}
Therefore, R is symmetric.

Non-Transitivity:
Suppose \mathrm{A}, \mathrm{B} and \mathrm{C} \in \mathrm{S}, such that (\mathrm{A}, \mathrm{B}) \in \mathrm{R} and (\mathrm{B}, \mathrm{C})
As a result, R is reflexive, symmetric but not transitive.

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