Let S be the set of all lines in 3 dimensional space. A relation ’R’ is defined on S by “l R m if
and only if l lies on the plane of m” for l, m ∈ S. Examine if R is (i) reflexive, (ii) symmetric,(iii)
transitive. Give your conclusion with graphical representation

Let \( S \) be the set of all lines in 3-dimensional space, and a relation \( R \) be defined on \( S \) such that:

- For \( l, m \in S \), \( l R m \) means "line \( l \) lies on the plane of line \( m \)."

We will now analyze whether the relation \( R \) is **reflexive**, **symmetric**, and **transitive**.

### (i) Reflexive

A relation \( R \) is reflexive if \( l R l \) holds for every \( l \in S \).

- **Interpretation**: A line \( l \) lies on the plane defined by itself.
  
Since a line lies on any plane that contains it, it trivially lies on its own plane. Thus, \( l R l \) is always true.

- **Conclusion**: \( R \) is **reflexive**.

### (ii) Symmetric

A relation \( R \) is symmetric if \( l R m \implies m R l \) for all \( l, m \in S \).

- **Interpretation**: If line \( l \) lies on the plane of line \( m \), does line \( m \) lie on the plane of line \( l \)?

In 3D space, if line \( l \) lies on the plane of line \( m \), it is not necessarily true that \( m \) lies on the plane of \( l \). For example, \( l \) could be on the plane that contains \( m \), but \( m \) could be skewed or otherwise positioned outside the plane of \( l \).

- **Conclusion**: \( R \) is **not symmetric**.

### (iii) Transitive

A relation \( R \) is transitive if \( l R m \) and \( m R n \) together imply \( l R n \) for all \( l, m, n \in S \).

- **Interpretation**: If line \( l \) lies on the plane of line \( m \) and line \( m \) lies on the plane of line \( n \), does line \( l \) lie on the plane of line \( n \)?

Consider this scenario in 3D space. If \( l \) lies on the plane of \( m \), and \( m \) lies on the plane of \( n \), it does not necessarily mean that \( l \) lies on the plane of \( n \). The planes of \( l \) and \( n \) may be different, so this condition does not hold in general.

- **Conclusion**: \( R \) is **not transitive**.

### Graphical Representation

To visualize this:

1. **Reflexive**: Draw a line \( l \), and show that it lies on its own plane.
2. **Symmetric**: Show two lines \( l \) and \( m \). Let \( l \) lie on the plane of \( m \), but \( m \) does not lie on the plane of \( l \).
3. **Transitive**: Consider three lines \( l, m, n \). Show \( l \) lying on the plane of \( m \), and \( m \) lying on the plane of \( n \), but \( l \) does not lie on the plane of \( n \).

Would you like me to generate a diagram of this representation for you?

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Here are 5 related questions:

1. What are other examples of reflexive but non-symmetric relations?
2. Can you provide a counter-example of symmetric but not transitive relations?
3. How does this relation change if we restrict it to 2D space?
4. What is the geometric interpretation of transitivity in terms of planes and lines?
5. How does this problem extend to higher-dimensional spaces?

**Tip**: Visualizing relations geometrically can clarify abstract concepts, especially in higher-dimensional problems.

Let S be the set of all lines in 3-dimensional space. A relation ‘R’ is defined on S by “l R m if and only if l lies on the plane of m” for l, m ∈ S. Examine if R is (i) reflexive, (ii) symmetric, (iii) transitive. Give your conclusion with graphical representation.

Analyze the relation R defined on the set of lines in 3D space, where l R m if line l lies on the plane of line m. Determine if the relation is reflexive, symmetric, and transitive with a detailed graphical representation. Suitable for Grades 10-12.

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