Types Of Relations

Step 1: Understanding the Concept:
Number of relations from to is .
Step 2: Detailed Explanation:
Given . So .
Step 3: Final Answer:
Option (D).

Concept: A relation on a set is:
• Reflexive: if for all
• Symmetric: if
• Transitive: if

Step 1: Check Reflexivity
All elements are present in .
Hence, relation is reflexive.

Step 2: Check Symmetry
Each ordered pair is of the form , so its reverse is also present.
Hence, relation is symmetric.

Step 3: Check Transitivity
If and are in , then is also in .
Hence, relation is transitive. Final Conclusion:
The relation is reflexive, symmetric and transitive.

Step 1: Express in terms of .
We are given the equation: To find the inverse, we first solve for in terms of .
Step 2: Rearrange the equation to isolate .
Multiply both sides by : Now, subtract from both sides: Finally, divide by to solve for :
Step 3: Take the inverse tangent to find .
Now, apply the inverse tangent to both sides:
Step 4: Conclusion.
Thus, the inverse function is: