Abstract
I have no desire to include a rigorous introduction to the theory of sets in this book. Perhaps what follows will motivate the interested reader to learn this theory in a special course on mathematical logic. In any case, the common intuitive understanding of a set as an abstract “aggregate of elements” is enough for our purposes. Any set can be imagined geometrically as a collection of points, and we will often refer to the elements of a set as points. By definition, all the elements of a set are distinct. A set X may be considered as having been adequately defined as soon as one can say that a given item is or is not an element of X. If x is an element of a set X, we write x ∈ X. Two sets are equal if they consist of the same elements. There is a unique set containing no elements. It is called the empty set and is denoted by \(\varnothing \). For a finite set X, we write | X | for the total number of elements in X and call it the cardinality of X. A set X is called a subset of a set Y if each element x ∈ X also belongs to Y. In this case, we write X ⊂ Y. Note that \(\varnothing \) is a subset of every set, and every set is a subset of itself. A subset of a set X that is not equal to X is said to be proper .
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Notes
- 1.
Also called the Cartesian product of sets.
- 2.
A set is called countable if it is isomorphic to \(\mathbb{N}\). An infinite set not isomorphic to \(\mathbb{N}\) is called uncountable.
- 3.
This is a particular case of the generic Newton’s binomial theorem , which expands (1 + x)s with an arbitrary α. We will prove it in Sect. 1.2.
- 4.
The upper left-hand corner of each diagram should coincide with that of the rectangle. The empty diagram and the whole rectangle are allowed.
- 5.
Note that the equality | λ | = n = m 1 + 2m 2 + ⋯ + nm n forces many of the m i to vanish.
- 6.
They are skew-symmetric, i.e., they satisfy the condition x 1 ∼ x 2 & x 2 ∼ x 1 ⇒ x 1 = x 2; see Sect. 1.4 on p. 13.
- 7.
See Sect. 1.2 on p. 7.
- 8.
See Definition 1.1 on p. 8.
- 9.
Every such relation is called a partial preorder on \(\mathbb{Z}\).
- 10.
Also nondecreasing or nonstrictly increasing or a homomorphism of posets.
- 11.
Such an element is unique, as we have seen above.
- 12.
To be more precise (see Sect. 1.3.2 on p. 11), let \(I \subset \mathcal{W}\times P\) consist of all pairs (W, c) such that w < c for all w ∈ W. Then the projection \(\pi _{1}: I \rightarrow \mathcal{W}\), (W, c) ↦ W, is surjective, because by the assumption of the lemma, for every W, there exists some upper bound d, and then we have assumed that there exists some c > d. Take \(b: \mathcal{W}\rightarrow P\) to be the composition π 2 ∘ g, where \(g: \mathcal{W}\rightarrow I\) is any section of π 1 followed by the projection π 2: I → P, (W, c) ↦ c.
- 13.
The total degree of the monomial \(x_{1}^{m_{1}}x_{2}^{m_{2}}\cdots x_{n}^{m_{n}}\) equals ∑ i = 1 n m i .
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Gorodentsev, A.L. (2016). Set-Theoretic and Combinatorial Background. In: Algebra I. Springer, Cham. https://doi.org/10.1007/978-3-319-45285-2_1
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