Types Of Relations

To determine what kind of relation $R$ is over the set $A$ of all straight lines in a plane, consider the definition of the relation: $l_1%20%5C%2C%20R%20%5C%2C%20l_2%20%5Ciff%20l_1$ is parallel to $l_2$ . Let's analyze the properties of equivalence relations: reflexivity, symmetry, and transitivity.

Reflexivity: A relation $R$ on set $A$ is reflexive if every element is related to itself. For any line $l_1$ in $A$ , it is clear that $l_1%20%5Cparallel%20l_1$ because any line is parallel to itself. Hence, $R$ is reflexive.

Symmetry: A relation $R$ on set $A$ is symmetric if for any elements $l_1%2C%20l_2$ in $A$ , whenever $l_1%20%5C%2C%20R%20%5C%2C%20l_2$ , then $l_2%20%5C%2C%20R%20%5C%2C%20l_1$ . If $l_1$ is parallel to $l_2$ , then $l_2$ is parallel to $l_1$ . Thus, $R$ is symmetric.

Transitivity: A relation $R$ on set $A$ is transitive if for any elements $l_1%2C%20l_2%2C%20l_3$ in $A$ , whenever $l_1%20%5C%2C%20R%20%5C%2C%20l_2$ and $l_2%20%5C%2C%20R%20%5C%2C%20l_3$ , then $l_1%20%5C%2C%20R%20%5C%2C%20l_3$ . If $l_1%20%5Cparallel%20l_2$ and $l_2%20%5Cparallel%20l_3$ , then $l_1%20%5Cparallel%20l_3$ . Thus, $R$ is transitive.

Since $R$ is reflexive, symmetric, and transitive, it is an equivalence relation.

High-Yield Trend

Chapter Questions
1 MCQs

Official Solution

Types Of Relations

High-Yield Trend

Chapter Questions 1 MCQs

Official Solution

Chapter Questions
1 MCQs