Summation of Unity over Elements

Theorem

Let $S \subseteq \Z$ be a set of integers.

Let:

$n := \ds \sum_{j \mathop \in S} 1$

Then $n$ is equal to the cardinality of $S$.

Proof

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Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercise $17$