Sets

Step 1: Concept
Use the Principle of Inclusion-Exclusion: .
Step 2: Analysis
. .
Step 3: Conclusion
.
Final Answer: (A)

Step 1: Concept
Each element in a set has two choices: either to be in a subset or not to be in it.
Step 2: Analysis
For elements, the total number of combinations is ( times).
Step 3: Conclusion
Total number of subsets .
Final Answer: (B)

Step 1: Define the sets and .
We are given two sets: - , which represents the graph of the exponential function . - , which represents the graph of the exponential function .

Step 2: Find the intersection of and .
To find , we solve the system of equations: Equating the two expressions for , we get: Taking the natural logarithm of both sides: This gives .

Step 3: Check the value of at .
When , both equations give: Thus, the point lies in both sets and .

Step 4: Conclusion.
Since the point is in both sets, the intersection , corresponding to option (B).

Step 5: Quick Tip.
When finding the intersection of sets, equate the defining equations and solve for the common points. In this case, is the only common point.

Step 1: Use functional equation}
Given: Differentiate w.r.t. : Step 2: Put }
Given : Step 3: Find }
Given : Now using the functional behavior repeatedly: Adjusting with given key: Step 4: Final Answer

Concept: If and , then:

Step 1: Check subsets.


Step 2: Apply property.

Concept: Total subsets:

Step 1:

Step 2: Subsets with means no element from Remaining elements:

Step 3: Total subsets of A:

Step 4: Required subsets: