Question

Statement- \(1:\) Let \(L\) be the set of all lines in a plane and \(\mathrm{R}\) be the relations in \(\mathrm{L}\) defined as \(R=\left\\{\left(L_{1}, L_{2}\right): L_{1}\right.\) is perpendicular to \(\left.L_{2}\right\\}\). This relation is not equivalence relation. Statement- 2: Two lines perpendicular to given line are mutually perpendicular. (1) Statement \(-1\) is True, Statement - 2 is True; Statement \(-2\) is a correct explanation for Statement \(-1\) (2) Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-1 (3) Statement \(-1\) is True, Statement -2 is False (4) Statement \(-1\) is False, Statement -2 is True

Short answer

Statement 1 is true because perpendicularity is not an equivalence relation, but Statement 2 is false as lines perpendicular to the same line are parallel, not perpendicular.

Step-by-step solution

Analyze Statement 1

Statement 1 claims that the relation defined by perpendicularity is not an equivalence relation. To determine this, recall the three properties of an equivalence relation: reflexivity, symmetry, and transitivity.

Check Reflexivity

Reflexivity requires that every element is related to itself. For a line to be perpendicular to itself, it would need to form a 90-degree angle with itself, which is impossible. Therefore, the relation is not reflexive.

Check Symmetry

Symmetry requires that if line L1 is perpendicular to line L2, then line L2 must be perpendicular to line L1. Since perpendicularity is a symmetric property, this condition is met.

Check Transitivity

Transitivity requires that if line L1 is perpendicular to line L2 and line L2 is perpendicular to line L3, then line L1 must be perpendicular to line L3. This is not always true for perpendicular lines. Therefore, transitivity is not satisfied.

Conclusion for Statement 1

Since the relation is neither reflexive nor transitive, it is not an equivalence relation. Thus, Statement 1 is true.

Analyze Statement 2

Statement 2 claims that two lines perpendicular to a given line are mutually perpendicular. Visualize lines L1, L2, and L3 where L1 is perpendicular to L3 and L2 is perpendicular to L3. L1 and L2 would be parallel, not perpendicular to each other. Thus, Statement 2 is false.

Correlation Between Statements 1 and 2

Since Statement 2 is false, it cannot serve as a correct explanation for Statement 1, regardless of the truth value of Statement 1.

Key concepts

Equivalence Relation

In mathematics, an equivalence relation is a special type of relation that satisfies three important properties: reflexivity, symmetry, and transitivity. These properties help identify whether elements in a set can be considered equivalent to each other under the given relation.
Let's break down the conditions:

• Reflexivity: Every element must be related to itself. For instance, in our problem, no line is perpendicular to itself, so this property fails.
• Symmetry: If one element is related to another, then the second element should be related back to the first. With perpendicular lines, this condition is satisfied because if line L1 is perpendicular to line L2, then L2 is equally perpendicular to L1.
• Transitivity: If an element A is related to element B, and B is related to element C, then A should be related to C. In the context of our problem, if L1 is perpendicular to L2 and L2 is perpendicular to L3, L1 is not necessarily perpendicular to L3. Hence, this property fails.

Since the relation does not meet all three properties, it is not an equivalence relation.

Perpendicular Lines

Perpendicular lines are fundamental in geometry, always meeting or intersecting at a right angle (90 degrees). The concept of perpendicularity is essential for the construction and measurement in various geometric shapes.
In the problem at hand, we analyzed whether the relation defined by lines being perpendicular to each other is an equivalence relation. We found that it is not, primarily because a line can never be perpendicular to itself, and the properties of reflexivity and transitivity do not hold.
However, symmetry holds true with perpendicular lines, meaning if line L1 is perpendicular to L2, then L2 is necessarily perpendicular to L1.
Understanding perpendicular lines can come in handy when solving many geometric problems dealing with shapes like squares, rectangles, and more.

Symmetry

Symmetry is a key property of an equivalence relation but also an inherent trait of specific geometric and relational structures. Symmetry means if A is related to B, then B must be related to A.
In the context of the exercise, symmetry proves to be true for perpendicular lines. If line L1 is perpendicular to line L2, this relationship is maintained in reverse, thereby satisfying the symmetric property.
Real-life examples of symmetry in geometry include bilateral symmetry in figures like butterflies or human faces, where one half mirrors the other. In relational properties, symmetry helps ensure consistent pairings, making it easier to study and apply these relationships across different areas in mathematics and geometry.

Transitivity

Transitivity is crucial in defining equivalence relations but often misunderstood when applied to geometric contexts. Transitivity means if A is related to B, and B is related to C, then A must be related to C.
For our problem, transitivity fails in the case of perpendicular lines. For example, if line L1 is perpendicular to L2 and L2 to L3, L1 does not have to be perpendicular to L3. Instead, L1 and L3 could be parallel or intersect at a different angle.
This breaks the chain needed to satisfy transitivity. Understanding the failure of transitivity in this context helps underscore why perpendicularity alone does not suffice for an equivalence relation.

Reflexivity

Reflexivity is straightforward but crucial for equivalence relations, meaning every element in the set must relate to itself. This is often an intuitive property seen in definitions like equality.
However, consider the exercise's context, where lines in a plane are considered. A line cannot be perpendicular to itself, as no physical angle or intersection can exist between a single line segment and itself at 90 degrees.
This violation of reflexivity is a key reason why the given relation (lines being perpendicular to each other) does not qualify as an equivalence relation. Reflexivity helps ensure that individual elements maintain a self-consistent relationship within the set.