$X$ 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:

Question

The following question appeared in JEE 2005.

Question:

$X$ 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

• A) $f(f^{-1}(b))=b$
• B) $f^{-1}(f(a))=a$
• C) $f(f^{-1}(b))=b, b\subset Y$
• D) $f^{-1}(f(a))=a, a\subset X$

My Attempt:

$c$ is a subset of $X$, $y$ is a subset of $Y$. $a$ is a subset of $x$. The shared link has drawn a pic of this situation.

Also, $d$ is a subset of $Y$, $x$ is a subset of $X$.

Since $f^{-1}(d)=x$, we may say $f(x)=d$

And since $a\subset x$, therefore, $f(a)\subset d$

So, option D) is correct. And the similar reasoning can be applied on option C)

But the answer given is only D)

Why is option C) incorrect?