When checking if a relation is an equivalence relation, remember to test for reflexivity, symmetry, and transitivity:
1. Reflexivity: Ensure every element is related to itself (e.g., a line is always parallel to itself).
2. Symmetry: Check if the relation works in both directions (e.g., if one line is parallel to another, the reverse is true).
3. Transitivity: Verify that if the relation holds between two pairs, it must also hold between the first and last element (e.g., if one line is parallel to a second, and the second is parallel to a third, the first is parallel to the third).
If all three properties are satisfied, the relation is an equivalence relation, dividing the set into equivalence classes.
To classify the relation \( R \) on the set \( A \) of all planar straight lines, defined by \( l_1 \, R \, l_2 \iff l_1 \) is parallel to \( l_2 \), we examine its properties as an equivalence relation: reflexivity, symmetry, and transitivity.
Reflexivity: A relation \( R \) on set \( A \) is reflexive if every element \( l_1 \in A \) satisfies \( l_1 \, R \, l_1 \). As any line \( l_1 \) is parallel to itself (\( l_1 \parallel l_1 \)), the relation \( R \) is reflexive.
Symmetry: A relation \( R \) on set \( A \) is symmetric if for any \( l_1, l_2 \in A \), \( l_1 \, R \, l_2 \) implies \( l_2 \, R \, l_1 \). If \( l_1 \) is parallel to \( l_2 \), it follows that \( l_2 \) is parallel to \( l_1 \). Therefore, \( R \) is symmetric.
Transitivity: A relation \( R \) on set \( A \) is transitive if for any \( l_1, l_2, l_3 \in A \), \( (l_1 \, R \, l_2 \text{ and } l_2 \, R \, l_3) \) implies \( l_1 \, R \, l_3 \). If \( l_1 \parallel l_2 \) and \( l_2 \parallel l_3 \), then \( l_1 \parallel l_3 \). Consequently, \( R \) is transitive.
Because \( R \) satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation.