# Unary Operation/Examples/All Mappings are Unary

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## Example of Unary Operation

To a set theorist, all mappings are unary operations.

## Proof

Let $\ds \prod_{i \mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$.

Let $f: \ds \prod_{i \mathop = 1}^n S_i \to T$ be a mapping from $\prod_{i \mathop = 1}^n S_i$ to $T$.

Thus:

- $f \subseteq \paren {S_1 \times S_2 \times \dotsb \times S_n} \times T$

Hence to consider $f$ as a unary operation, one would consider $\ds \prod_{i \mathop = 1}^n S_i$ as:

- a set whose elements are ordered $n$-tuples

as opposed to:

- an $n$-dimensional cartesian product of sets.

A set theorist, to a certain level of approximation, considers a set to be an aggregation of *any* objects.

Hence a set of ordered $n$-tuples is a convenient way to regard the domain of $f$.

$\blacksquare$

## Sources

- 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.6$: Functions: Exercise $1.6.1$