Preliminaries
This chapter aims to recall basic concepts of set theory and abstract algebra including set, relation, isomorphism, lattice, Boolean algebra and soft algebra, which will serve as the base of the remaining chapters in the book.
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Preliminaries
This chapter aims to recall basic concepts of set theory and abstract algebra including set, relation, isomorphism, lattice, Boolean algebra and soft algebra, which will serve as the base of the remaining chapters in the book.
1.1
Sets
A set is viewed as a collection of objects satisfying certain desired properties. Every object in the set is mathematically called an element or member. If every element of set A is also an element of set B, then A is called a subset of B and this is written as A ⊆ B. If A ⊆ B and B ⊆ A, then we say that A and B are equal, written as A = B. If A ⊆ B, A = B and A = ∅, where ∅ denotes the empty set, then A is called a proper subset of B, denoted by A ⊂ B. Let X be the universe of discourse (or the universe for short). The set of all subsets of X is denoted by P (X), i.e. P (X) = {A|A ⊆ X}, and is called the power set (class) of X. For A, B ∈ P (X), some of the set-theoretic operations are defined as follows: The union of A and B is defined by A ∪ B = {x|x ∈ A or x ∈ B}. The intersection of A and B is defined by A ∩ B = {x|x ∈ A and x ∈ B}. The complement of A is defined by Ac = {x|x ∈ X and x ∈ A}. Proposition 1.1. The above defined set-theoretic operations satisfy (∀A, B, C ∈ P (X)): (1) idempotency: A ∪ A = A, A ∩ A = A; (2) commutativity: A ∪ B = B ∪ A, A ∩ B = B ∩ A; X. Wang et al.: Mathematics of Fuzziness – Basic Issues, STUDFUZZ 245, pp. 1–20. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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1 Preliminaries
(3) associativity: (A ∪ B) ∪ C = A ∪ (B ∪ C), (A ∩ B) ∩ C = A ∩ (B ∩ C); (4) absorption laws: A ∪ (A ∩ B) = A, A ∩ (A ∪ B) = A; (5) distributivity: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C); (6) the existence of the greatest and least element: ∅ ⊆ A ⊆ X. (7) involution: (Ac )c = A; (8) De Morgan laws: (A ∪ B)c = Ac ∩ B c , (A ∩ B)c = Ac ∪ B c ; (9) complementation: A ∪ Ac = X(the law of excluded middle), A ∩ Ac = ∅ (the law of contradiction). Proof. We only prove the first equality of (8) as an illustration. The remaining proofs are left to the reader. ∀x ∈ X, x ∈ (A ∪ B)c ⇔ x ∈ A ∪ B ⇔ (x ∈ A and x ∈ B) ⇔ x ∈ Ac ∩ B c . More generally, the union and intersection of Ai (i ∈ I) may be similarly defined with an arbitrary index set I. Ai = {x|∃i ∈ I, x ∈ Ai }; i∈I
Ai = {x|∀i ∈ I, x ∈ Ai }.
i∈I
Accordingly, the associativity, distributivity and De Morgan laws are extended as (Ai ∩ Bi ), ( Ai ) ∪ ( Bi ) = (Ai ∪ Bi ) and ( Ai ) ∩ ( Bi ) = i∈I
A∪( (
i∈I
i∈I
Ai ) =
i∈I
Ai )c =
i∈I
i∈I
i∈I
(A ∪ Bi ) and A ∩ (
i∈I
(Ai )c and (
Bi ) =
i∈I
Ai )c =
i∈I
i∈I
i∈I
(A ∩ Bi ),
i∈I
(Ai )c ,
i∈I
where I is an arbitrary index set and A, Ai , Bi ∈ P (X)(∀i ∈ I). In describing sets, an important tool is the characteristic function. Let A be a subset of X. The characteristic function of A is defined by: ∀x ∈ X, 1 x ∈ A; χA (x) = 0 otherwise. The set-theoretic operations of union, intersection and complement may be expressed by means of characteristic functions. Proposition 1.2.
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