Data structure

Anmerkungen:

• characteristics of data structures
• are sets of variables, of different types connected together in various ways and with a set of defined operations on such structures.

1. Some examples found in math
   1. algebraic structures
      1. groups
      2. rings
      3. bodies
   2. discrete structures
      1. trees
      2. automata
      3. graphs
2. Operations performed on a data structure
   1. Browse by structure

      Anmerkungen:

      • Make a tour of the structure in order to retrieve the stored information. 
   2. Search

      Anmerkungen:

      • Determines whether an item is or is not in the structure.
   3. Total or partial copy

      Anmerkungen:

      • Acquired wholly or partly with a structure similar to the original features. 
   4. Information query

      Anmerkungen:

      • Get information of one or more elements of the structure 
   5. Test

      Anmerkungen:

      • Check if one or more elements meet certain conditions
   6. Modification

      Anmerkungen:

      • Varies partially or totally the contents of the information elements of the structure 
   7. Print

      Anmerkungen:

      • Enter the information contained in the structure.
   8. Elimination

      Anmerkungen:

      • Deletes elements of the structure 
3. Basic functions for creation, access and destruction of Data Structures
   1. Constructor function

      Anmerkungen:

      • Create the structure, ie, define the characteristics, delimitation, relationships and allocate the space, leaving to the user the structure to proceed to place the information. 
   2. Function to access

      Anmerkungen:

      • Facilitate the arrival of an element belonging to the structure. 
   3. Destructor function

      Anmerkungen:

      • Return to system resources allocated to the data structure so that they are available to the payee
4. Data

   Anmerkungen:

   • Are associated with qualities or quantities and events or objects that are processed by the computer
   1. Common data types
      1. Qualitative
      2. Crisp

         Anmerkungen:

         • not possible to describe phenomena that manifest some imprecision and / or uncertainty, both in representation and an inquiry 
         1. Examples
            1. Numerical
            2. Alphanumeric
            3. Binary
      3. Diffuse.

         Anmerkungen:

         • formalizes vague or fuzzy concepts that people handled daily and naturally.
5. Fuzzy Logic
   1. Finite logics
      1. Examples

         Anmerkungen:

         • • Irrational number π. • The sequence of rational numbers are approximate values ​​of π. • Closed interval [3, π] as domain approach. • The order of approximation required.
   2. Fuzzy logic

      Anmerkungen:

      • It is a pair (U, g), where U is the universe of discourse of the propositions diffuse, and g: U → R is a real function of propositional variable, being I the closed interval [0, 1]. In this case, we say that g is veritative fuzzy logic function (U, g).
   3. Standard fuzzy logic
      1. Properties
         1. Commutativity of ∨

            Anmerkungen:

            • (p ∨ q) ≡ (q ∨ p)
         2. Commutativity of ∧

            Anmerkungen:

            • p ∧ q) ≡ (q ∧ p) 
         3. Associativity of ∨

            Anmerkungen:

            • ((p ∨ q) ∨ r) ≡ (p ∨ (q ∨ r))
         4. Associativity of ∧

            Anmerkungen:

            • ((p ∧ q) ∧ r) ≡ (p ∧ (q ∧ r))
         5. Distributivity (∨, ∧)

            Anmerkungen:

            • (p ∨ (q ∧ r)) ≡ ((p ∨ q) ∧ (p ∨ r))
         6. Neutral element of ∨

            Anmerkungen:

            • (p ∨ 0) ≡ p
         7. Distributivity (∧, ∨)

            Anmerkungen:

            • (p ∧ (q ∨ r)) ≡ ((p ∧ q) ∨ (p ∧ r))
         8. 2nd Law of De Morgan

            Anmerkungen:

            • [ ∼ (p ∨ q) ] ≡ (∼p ∧ ∼q)
         9. Neutral element of ∧

            Anmerkungen:

            • (p ∧ 1) ≡ p 
         10. Absorption (∧, ∨)

             Anmerkungen:

             • (p ∧ (p ∨ q)) ≡ p
         11. 1st Law of De Morgan

             Anmerkungen:

             • [∼ (p ∧ q) ] ≡ (∼p ∨ ∼q)
         12. Absorption (∨, ∧)

             Anmerkungen:

             • (p ∨ (p ∧ q)) ≡ p
      2. Operations
         1. Denial
         2. Conjunction
         3. Disjunction
         4. Biconditional
         5. Implication
   4. Fuzzy predicates unit

      Anmerkungen:

      • We call fuzzy predicate unit in the universe X to a function propositional P: X → U, that is, to correspond to each element x ∈ X, associates a unique proposition P (x) ∈ U
      1. Examples
         1. The sentence "x

            Anmerkungen:

            • It is a rational number irrational number approximately equal to π "variable Single xy individual constant π, is the specification of a fuzzy predicate P Unit set Q of all rational numbers in the universe of discourse of the fuzzy propositions U. Such specification is denoted by the expression P (x) "x ≈ π ". The degrees of truth of propositions P (3), P (3.1), P (3.14), P (3,141), P (3.1415), ..., constitute a series, infinite and growing of rational numbers between 0 and 1.
         2. The statement "x

            Anmerkungen:

            • It is a rational number irrational number approximately equal to π "variable Single xy individual constant π, is the specification of a fuzzy predicate P Unit set Q of all rational numbers in the universe of discourse of the fuzzy propositions U. Such specification is denoted by the expression P (x) "x ≈ π ". The degrees of truth of propositions P (3), P (3.1), P (3.14), P (3,141), P (3.1415), ..., constitute a series, infinite and growing of rational numbers between 0 and 1.
   5. Fuzzy proposition p
6. Fuzzy Sets

   Anmerkungen:

   • There are sets in which is not clearly determined whether an element belongs or not to the set. Sometimes, an element in the set with a certain degree, called membership degree.
   1. Finite fuzzy set

      Anmerkungen:

      • A = {a1 | uA (a1), a2 | uA (a2), a3 | uA (a3), ..., an | uA (n)}
   2. Countably infinite fuzzy set

      Anmerkungen:

      • B = {b1|µB(b1), b2|µB(b2), b3|µB(b3), …}
   3. Relations between fuzzy sets
      1. Equality

         Anmerkungen:

         • The fuzzy set A is equal to the fuzzy set B if and only if for each element x ∈ X, the number uA (x) equals the number μB (x). We denote A = B. In compact form: A = B ⇔ (∀ x ∈ X) [uA (x) = μB (x)].
      2. Inclusion

         Anmerkungen:

         • The fuzzy set A is a subset of the fuzzy set B if and only if for all x ∈ X, the number uA (x) is less than or equal to the number μB (x). Denoted A ⊂ B. In compact form: A ⊂ B ⇔ (∀ x ∈ X) [uA (x) ≤ μB (x)].
   4. Operations with fuzzy sets
      1. Complement

         Anmerkungen:

         • The complement of a fuzzy set A is a second fuzzy set B of generic element x ∈ X such that μB (x) equals the number number 1 - uA (x). Denoted Ac. In compact form: Ac = B ⇔ (∀ x ∈ X) [μB (x) = 1 - uA (x)].
      2. Union

         Anmerkungen:

         • The union of two fuzzy sets A and B is another fuzzy set C of generic element x ∈ X such that μC (x) equals the maximum between uA (x) and μB (x) numbers. It is denoted A ∪ B. In compact form: A ∪ B = C ⇔ (∀ x ∈ X) μC (x) = max {uA (x), μB (x)}.
      3. Difference

         Anmerkungen:

         • The difference of two fuzzy sets A and B is a third fuzzy set, denoted by A-B, and defined by the following equality: A - B = A ∩ Bc 
      4. Symmetric difference

         Anmerkungen:

         • The symmetric difference of two fuzzy sets A and B is a third fuzzy set AΔB denoted by, and defined as follows: AΔB = (A ∩ Bc) ∪ (Ac ∩ B).
      5. Intersection

         Anmerkungen:

         • The intersection of two fuzzy sets A and B is another fuzzy set C of generic element x ∈ X such that μC (x) is equal to the minimum between uA (x) and μB (x) numbers. It is denoted A ∩ B. In compact form: A ∩ B = C ⇔ (∀ x ∈ X) μC (x) = min {uA (x), μB (x)}.