Question

Let \( R \) be the relation over the set \( A \) of all straight lines in a plane such that \( l_1 \, R \, l_2 \iff l_1 \) is parallel to \( l_2 \). Then \( R \) is:

• A. Symmetric
• B. An Equivalence relation
• C. Transitive
• D. Reflexive

Hint:

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.

Answer 1

The Correct Option is B

The relation R is defined as l₁R l₂ ↔ l₁ is parallel to l₂. To check if R is an equivalence relation, we need to verify the following properties:

Reflexivity: A line is parallel to itself, so l₁R l₁ holds for all l₁, so the relation is reflexive.

Symmetry: If l₁ is parallel to l₂, then l₂ is parallel to l₁, so the relation is symmetric.

Transitivity: If l₁ is parallel to l₂, and l₂ is parallel to l₃, then l₁ is parallel to l₃, so the relation is transitive.

Since all three properties hold, the relation R is an equivalence relation.

Answer 2

The relation \( R \) is defined as \( l_1 R l_2 \) if and only if \( l_1 \) is parallel to \( l_2 \). To check if \( R \) is an equivalence relation, we need to verify three essential properties: reflexivity, symmetry, and transitivity. An equivalence relation divides the set into distinct equivalence classes, where each element is equivalent to every other element within the same class. We will go through these properties step-by-step to confirm if the relation \( R \) is indeed an equivalence relation:

Reflexivity: A relation is reflexive if every element in the set is related to itself. In this case, since a line is always parallel to itself, the relation \( l_1 R l_1 \) holds for any line \( l_1 \). Therefore, \( R \) is reflexive because any line is parallel to itself.

Symmetry: A relation is symmetric if for any two elements \( l_1 \) and \( l_2 \), if \( l_1 R l_2 \) holds, then \( l_2 R l_1 \) must also hold. In this case, if line \( l_1 \) is parallel to line \( l_2 \), then \( l_2 \) is also parallel to \( l_1 \). Therefore, the relation is symmetric, as parallelism between two lines is mutual.

Transitivity: A relation is transitive if for any three elements \( l_1 \), \( l_2 \), and \( l_3 \), if \( l_1 R l_2 \) and \( l_2 R l_3 \), then \( l_1 R l_3 \). In this case, if line \( l_1 \) is parallel to line \( l_2 \) and line \( l_2 \) is parallel to line \( l_3 \), then line \( l_1 \) must be parallel to line \( l_3 \). Thus, the relation is transitive because parallelism between lines is a transitive property.

Conclusion: Since the relation \( R \) is reflexive, symmetric, and transitive, it satisfies all the conditions to be an equivalence relation. Therefore, \( R \) is an equivalence relation, meaning that it groups lines into equivalence classes of parallel lines.