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Motivation: Usage of Ordered Tuples In Other Mathematical Disciplines
(related to Part: Relations)
We will see soon that, inside the set theory, relations and functions are ordered $n$-tuples. But the concept of ordered $n$-tuples is used in many other mathematical disciplines to define key concepts there, for instance:
- The algebraic structures are $n$-tuples of sets and binary operations defined on them. We will see later that binary operations are, in fact, sets. Therefore using $n$-tuples to describe them is reasonable.
- The same holds for key topological structures like the metric space.
- A vector of numbers is an $n$-tuple. When we talk about number systems, we will provide a set-theoretical foundation of numbers. This means that also numbers are nothing else but sets. Therefore, using $n$-tuples to describe vectors of numbers is also reasonable.
- A graph, the key concept of graph theory is an $n$-tuple of sets.
These connections to other mathematical disciplines and that they use the language of set theory to define their key concepts, is the main reason, why set theory is today considered the foundation of modern mathematics.