Cardinality of Set of Injections/Corollary
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Corollary to Cardinality of Set of Injections
Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Then $f$ cannot be an injection if:
- $\card S > \card T$
where $\card S$ denotes the cardinality of $S$.
Proof
From Cardinality of Set of Injections:
- the number of injections from $S$ to $T$, where $\card S > \card T$, is zero.
Hence the result.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Mappings: Exercise $4 \ \text{(ii) (b)}$
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Ponderable $2.1.5$