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

Let $f: \displaystyle \prod_{i \mathop = 1}^n S_i \to T$ 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 $\displaystyle \prod_{i \mathop = 1}^n S_i$ 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, a set of ordered $n$-tuples is a convenient way to regard the domain of $f$.

$\blacksquare$


Sources