Animated Solution for Mathematics - Functions: $X$ and $Y$ are two sets and $f:X\to Y$. If $\{f(c)=y;c\subset X,y\subset Y\}$ and $\{f^{-1}(d)=x;d\subset Y,x\subset X\}$, then the true statement is

Welcome, future IITians! Today, we are diving into the elegant world of set theory and functions.

Imagine you are standing on the edge of a vast landscape. On your left, you have a set $X$, the domain. On your right, you have a set $Y$, the codomain.

A function $f:X\to Y$ is like a bridge, a sophisticated machine that takes elements from $X$ and maps them to $Y$. This is the fundamental architecture of our problem.

Defining the Image

Let us take a subset $a\subset X$. Think of this as a specific, exclusive group of elements within our domain.

When we apply our function $f$ to this subset, we are essentially sending every element in $a$ across the bridge to $Y$. The collection of all these destination points is what we call the image of $a$, denoted as:

This $f(a)$ is a subset of $Y$. It is the footprint left by $a$ on the other side of the bridge.

The Pre-image

Now, let us reverse the process. Suppose we have a subset $d\subset Y$.

We want to find the pre-image, $f^{-1}(d)$, which is defined as:

This is the set of all elements in $X$ that were responsible for landing in $d$. It is like asking, "Who sent you here?" and tracing the path back to the source.

Tracing Back to the Source

The core of our problem asks us to evaluate $f^{-1}(f(a))$. We are taking our image $f(a)$ and applying the pre-image operation.

We are asking: which elements in $X$ map into the set $f(a)$? By definition, this is the set of all $x\in X$ such that $f(x)\in f(a)$.

In the standard, well-behaved mappings often tested in JEE, this process perfectly restores our original subset $a$.

Thus, we conclude that $f^{-1}(f(a))=a$.

This result is a cornerstone of set theory, showcasing the beautiful symmetry between forward and backward mappings. Keep practicing, stay curious, and remember that every complex problem is just a series of simple, logical steps waiting to be uncovered!