If X and Y are two non-empty sets, where `f:X rarr 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(f^(-1)(B))=B" only if B "subseteqf(X)`
D. `f(f^(-1)(B))=B`

Answer

C

Explanation

We have
`A subseteq f^(-1)(f(A)) and A=f^(-1)(f(A))` iff f is a bijection.
So, options (a) and (b) are not correct
`f(f^(-1)(B)) subseteq B and f(f^(-1)(B))=B" iff "B subseteqf(X)`
So, options ( c) is correct but options (d) is not correct.