Data structure

Annotations:

• 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

      Annotations:

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

      Annotations:

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

      Annotations:

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

      Annotations:

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

      Annotations:

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

      Annotations:

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

      Annotations:

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

      Annotations:

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

      Annotations:

      • 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

      Annotations:

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

      Annotations:

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

   Annotations:

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

         Annotations:

         • 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.

         Annotations:

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

         Annotations:

         • • 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

      Annotations:

      • 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 ∨

            Annotations:

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

            Annotations:

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

            Annotations:

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

            Annotations:

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

            Annotations:

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

            Annotations:

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

            Annotations:

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

            Annotations:

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

            Annotations:

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

             Annotations:

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

             Annotations:

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

             Annotations:

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

      Annotations:

      • 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

            Annotations:

            • 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

            Annotations:

            • 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

   Annotations:

   • 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

      Annotations:

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

      Annotations:

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

         Annotations:

         • 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

         Annotations:

         • 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

         Annotations:

         • 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

         Annotations:

         • 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

         Annotations:

         • 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

         Annotations:

         • 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

         Annotations:

         • 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)}.