Part BCSIR NET June 2025the-set-of-maps-is-far-bigger-than-the-sets-it-runs-between

The set of maps is far bigger than the sets it runs between

Let A, B be non-empty subsets of $\mathbb{N}$ with cardinality |A| ≥ 2. Let $S_{1} =$ {f : A → B | f is one-to-one} and $S_{2} =$ {g : B → A | g is onto}. Which of the following statements is true?

A.If A ⊊ B and B is finite, then there is a one-to-one map from $S_{2}$ to $S_{1}$ .
B.If $B = \mathbb{N}$ , then there exists a one-to-one map from $S_{2}$ to B.
C.If $B = \mathbb{N}$ and A is finite, then there exists a one-to-one map from B to $S_{1}$ .✓ Correct
D.If A is finite, then $S_{2}$ is finite for any B.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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The set of maps is far bigger than the sets it runs between

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