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A finite set

\mathcal{S}

of positive integers is called cardinal if

\mathcal{S}

contains the integer

|\mathcal{S}|

where

|\mathcal{S}|

denotes the number of distinct elements in

\mathcal{S}

. Let

f

be a function from the set of positive integers to itself such that for any cardinal set

\mathcal{S}

, the set

f(\mathcal{S})

is also cardinal. Here

f(\mathcal{S})

denotes the set of all integers that can be expressed as

f(a)

where

a \in \mathcal{S}

. Find all possible values of

f(2024)

Proposed by Sutanay Bhattacharya

Problem Statement