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.


Let's examine the given relation \( R \) on the set \( S \) of all lines in 3-dimensional space, where \( l R m \) means "line \( l \) lies on the plane of line \( m \)." We are tasked with determining whether this relation is reflexive, symmetric, and transitive.

### 1. Reflexivity
A relation is **reflexive** if for every element \( l \in S \), \( l R l \) holds true. In this case, for \( l R l \), the line \( l \) would have to lie on its own plane. 

- Any line lies on the plane that contains the line itself.
- Therefore, \( l R l \) is true for any line \( l \in S \).

**Conclusion**: The relation \( R \) is **reflexive**.

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### 2. Symmetry
A relation is **symmetric** if for any two lines \( l, m \in S \), if \( l R m \) holds, then \( m R l \) must also hold. In this case, if line \( l \) lies on the plane of line \( m \), does line \( m \) also lie on the plane of line \( l \)?

- It is possible for a line \( l \) to lie on the plane of line \( m \), but this does not necessarily mean that line \( m \) lies on the same plane as line \( l \). 
  - For example, if line \( l \) is contained within the plane that contains line \( m \), this doesn’t imply the reverse.
  - Line \( m \) may be skew to line \( l \), meaning they do not share the same plane.

**Conclusion**: The relation \( R \) is **not symmetric**.

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### 3. Transitivity
A relation is **transitive** if for any lines \( l, m, n \in S \), if \( l R m \) and \( m R n \), then \( l R n \) must hold. In this case, if line \( l \) lies on the plane of line \( m \), and line \( m \) lies on the plane of line \( n \), does it follow that line \( l \) lies on the plane of line \( n \)?

- Consider the following: Line \( l \) could lie on the plane containing line \( m \), and line \( m \) could lie on a plane containing line \( n \), but this does not necessarily imply that line \( l \) lies on the plane of line \( n \). 
  - The planes containing lines \( l \) and \( n \) could be different, even if both intersect with the plane of line \( m \).
  - For example, three mutually skew lines can exist in 3D space such that line \( l \) lies on the plane of \( m \), and \( m \) lies on the plane of \( n \), but line \( l \) does not lie on the plane of \( n \).

**Conclusion**: The relation \( R \) is **not transitive**.

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### Final Conclusion:
- The relation \( R \) is **reflexive**.
- The relation \( R \) is **not symmetric**.
- The relation \( R \) is **not transitive**.

Would you like any further details or clarifications?

Here are 5 related questions to expand on this concept:
1. What would change if the relation required two lines to lie in the same plane?
2. How would the relation change if we defined \( l R m \) as lines being parallel instead?
3. Can we define a similar relation for points and lines in 3D space?
4. What are some geometric interpretations of transitivity in different types of spaces (2D, 3D, etc.)?
5. How would this relation be different in a higher-dimensional space (4D or beyond)?

**Tip**: When analyzing geometric relations like this, sketching examples in 3D space can help visualize abstract concepts like symmetry and transitivity.

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.

Explore the geometric relation R defined on lines in 3-dimensional space, where l R m if line l lies on the plane of line m. Determine whether this relation is reflexive, symmetric, and transitive. This problem is ideal for students in Grades 11-12, covering key concepts in geometry and relations.

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