Abstract
All further discussion will be based of a formal definition of relation, given in Definition 7.1. Then, in Definition 7.2, we introduce the central notion of definability in structures, and we proceed with examples of structures with very small domains, including the two element algebraic field F 2 presented in the last section.
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Notes
- 1.
In mathematics, a structure is usually defined as a set with a sequence of relations on it. That is an additional feature that allows to use the same relation as an interpretation for different relation symbols. We will not discuss such structures in this book, hence our definition is a bit simpler.
- 2.
Technically, the set of relations of a structure is never empty, since the equality relation symbol = is included in the first-order language and is interpreted in each structure as equality. Hence, to be precise, a trivial structure is a structure with one relation only—the equality relation.
- 3.
The Wikipedia entry on ordered pairs has a list of other definitions and an informative discussion.
- 4.
There is a small issue here. If a is negative, then 2a is actually smaller than a, for example 2 ⋅ (−1) = −2, and − 2 is smaller than − 1 in the usual ordering of the real numbers. To make the definition of R more precise, we may say that b is “twice larger” than a if a and b are both positive, or both negative, and the absolute value of b is twice the absolute value of a.
- 5.
One can define relation between elements of a set A and elements of a set B as an arbitrary subset of the Cartesian product A × B, or in greater generality, a relation can be defined as an arbitrary subset of the Cartesian product of a finite number of sets A 1 × A 2 ×⋯ × A n.
- 6.
A binary relation R is symmetric if for all a and b in the domain (a, b) ∈ R if and only if (b, a) ∈ R.
- 7.
This theorem is proved in Appendix A.3.
- 8.
In elementary algebra this formula is usually written as (a + b)2 = a 2 + 2ab + b 2.
- 9.
Instead of a 1 = a 1, we could use a i = a i, for any i, or in fact a i = a j for any i and j.
References
Tarski, A. (1948). A decision method for elementary algebra and geometry. Santa Monica: RAND Corporation.
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Kossak, R. (2018). Relations. In: Mathematical Logic. Springer Graduate Texts in Philosophy, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-97298-5_7
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