Number of Injective Restrictions

Theorem

Let $f: S \to T$ be a mapping.

Let $Q$ be the set of all injective restrictions of $f$.

Then the cardinality of $Q$ is:

$\ds \card {\prod_{i \mathop \in I} \prod_{j \mathop \in J_i} \family {\family {\powerset {S / \RR_f} }_i}_j}$

where:

$\powerset \cdot$ denotes power set

$S / \RR_f$ denotes quotient set of the induced equivalence of $f$.

Proof

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