Relations And Geometry

To determine the truth of the statements regarding the relation $R$ defined on $X%20%3D%20%5Cmathbb%7BR%7D%20%5Ctimes%20%5Cmathbb%7BR%7D$ , let's analyze each statement:

Statement-I: $R$ is an equivalence relation.
- Reflexivity: For any pair $(a%2C%20b)%20%5Cin%20X$ , we need $(a%2C%20b)%20%5C%2C%20R%20%5C%2C%20(a%2C%20b)$ . According to the definition of $R$ , this means $b%20%3D%20b$ , which is true. Therefore, $R$ is reflexive.
- Symmetry: For any $(a_1%2C%20b_1)%2C%20(a_2%2C%20b_2)%20%5Cin%20X$ , if $(a_1%2C%20b_1)%20%5C%2C%20R%20%5C%2C%20(a_2%2C%20b_2)$ , then by definition $b_1%20%3D%20b_2$ . For symmetry, we need $(a_2%2C%20b_2)%20%5C%2C%20R%20%5C%2C%20(a_1%2C%20b_1)$ , which also requires $b_2%20%3D%20b_1$ . Since equality is symmetric, $R$ is symmetric.
- Transitivity: For any $(a_1%2C%20b_1)%2C%20(a_2%2C%20b_2)%2C%20(a_3%2C%20b_3)%20%5Cin%20X$ , if $(a_1%2C%20b_1)%20%5C%2C%20R%20%5C%2C%20(a_2%2C%20b_2)$ and $(a_2%2C%20b_2)%20%5C%2C%20R%20%5C%2C%20(a_3%2C%20b_3)$ , this implies $b_1%20%3D%20b_2$ and $b_2%20%3D%20b_3$ . Therefore, $b_1%20%3D%20b_3$ , ensuring that $(a_1%2C%20b_1)%20%5C%2C%20R%20%5C%2C%20(a_3%2C%20b_3)$ , so $R$ is transitive.
Statement-II: For some $(a%2C%20b)%20%5Cin%20X$ , the set $S%20%3D%20%5C%7B(x%2C%20y)%20%5Cin%20X%20%3A%20(x%2C%20y)%20%5C%2C%20R%20%5C%2C%20(a%2C%20b)%5C%7D$ represents a line parallel to $y%20%3D%20x$ .

Conclusion: From the analysis above, we conclude that Statement-I is true but Statement-II is false.

High-Yield Trend

Chapter Questions
1 MCQs

Official Solution

Relations And Geometry

High-Yield Trend

Chapter Questions 1 MCQs

Official Solution

Chapter Questions
1 MCQs