If `X and Y` are two non-empty sets where `f: X->Y`,is function is defined such that `f(c) = {f (x): x in C}` for `C sube X and f^-1 (D) = {x: f(x) in D}` for `D sube Y`,for any `A sube Y and B sube Y`, then

Options

A. `f^(-1){f(A)}=A`
B. `f^(-1){f(A)}=A`, only if f(X)=Y
C. `f^(-1){f(B)}=B`, only if B `sube` f(x)
D. `f^(-1){f(B)}=B`

Answer

To solve the problem, we need to understand the definitions and properties of functions, particularly the concept of the inverse function. Let's break it down step by step.

### Step 1: Understanding the Function and Its Inverse
Given a function \( f: X \to Y \), we know that for any subset \( C \subseteq X \), the image of \( C \) under \( f \) is defined as:
\[ f(C) = \{ f(x) : x \in C \} \]
Similarly, for any subset \( D \subseteq Y \), the pre-image (or inverse image) of \( D \) under \( f \) is defined as:
\[ f^{-1}(D) = \{ x \in X : f(x) \in D \} \]

### Step 2: Analyzing the Given Subsets
Let \( A \) and \( B \) be subsets of \( Y \). We want to explore the relationships involving \( f \) and \( f^{-1} \) with respect to these subsets.

### Step 3: Finding \( f^{-1}(f(A)) \)
We can start by finding \( f^{-1}(f(A)) \):
- By definition, \( f(A) \) is the set of all images of elements in \( A \) under \( f \).
- The pre-image \( f^{-1}(f(A)) \) will give us all elements in \( X \) that map to any element in \( f(A) \).

Thus, we can conclude:
\[ f^{-1}(f(A)) \supseteq A \]
This means that the pre-image of the image of \( A \) under \( f \) contains \( A \).

### Step 4: Finding \( f(f^{-1}(B)) \)
Next, we find \( f(f^{-1}(B)) \):
- The set \( f^{-1}(B) \) consists of all elements in \( X \) that map to elements in \( B \).
- Applying \( f \) to this set gives us all images of those elements, which will be a subset of \( B \).

Thus, we conclude:
\[ f(f^{-1}(B)) \subseteq B \]

### Step 5: Conclusion
From the above steps, we can summarize the results:
1. \( f^{-1}(f(A)) \supseteq A \)
2. \( f(f^{-1}(B)) \subseteq B \)

These properties illustrate the relationships between a function and its inverse with respect to subsets of its codomain.