TS EAMCET · Maths · Functions

Let \(f: X \rightarrow Y\) be a function and \(A_y=\left\{\frac{f^{-1}(y)}{y \in Y}\right\}\). Then \(A_i \cap A_j=\dot{\phi}(i \neq j) \forall i, j \in Y\) and \(\bigcup_{y \in Y} A_y=X\), if

A \(f\) is onto function only
B \(f\) is one-one function only
C \(f\) is any function
D \(X\) and \(Y\) are finite sets only

Verified Solution

Answer & Solution

Correct Answer

(C) \(f\) is any function

Step-by-step Solution

Detailed explanation

It is given that function \(f: X \rightarrow Y\) and \(A_y=\left\{\frac{f^{-1}(y)}{y \in Y}\right\}\) \(\because\) Inverse of the function exists, so function \(f\) is one-one and onto. then \(A_i \cap A_j=\phi,(i \neq j) \forall i, j \in Y\), and \(\bigcup A_y=X\) for every…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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