Data structure

Data structure

1. Some examples found in math

   • algebraic structures
     • groups
     • rings
     • bodies
   • discrete structures
     • trees
     • automata
     • graphs

2. Operations performed on a data structure

   • Browse by structure
     • Make a tour of the structure in order to retrieve the stored information. 
   • Search
     • Determines whether an item is or is not in the structure.
   • Total or partial copy
     • Acquired wholly or partly with a structure similar to the original features. 
   • Information query
     • Get information of one or more elements of the structure 
   • Test
     • Check if one or more elements meet certain conditions
   • Modification
     • Varies partially or totally the contents of the information elements of the structure 
   • Print
     • Enter the information contained in the structure.
   • Elimination
     • Deletes elements of the structure 

3. Basic functions for creation, access and destruction of Data Structures

   • Constructor function
     • 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. 
   • Function to access
     • Facilitate the arrival of an element belonging to the structure. 
   • Destructor function
     • Return to system resources allocated to the data structure so that they are available to the payee

4. Data

   • Are associated with qualities or quantities and events or objects that are processed by the computer
   • Common data types
     • Qualitative
     • Crisp
       • not possible to describe phenomena that manifest some imprecision and / or uncertainty, both in representation and an inquiry 
       • Examples
         • Numerical
         • Alphanumeric
         • Binary
     • Diffuse.
       • formalizes vague or fuzzy concepts that people handled daily and naturally.

5. Fuzzy Logic

   • Finite logics
     • Examples
       • • 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.
   • Fuzzy logic
     • 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).
   • Standard fuzzy logic
     • Properties
       • Commutativity of ∨
         • (p ∨ q) ≡ (q ∨ p)
       • Commutativity of ∧
         • p ∧ q) ≡ (q ∧ p) 
       • Associativity of ∨
         • ((p ∨ q) ∨ r) ≡ (p ∨ (q ∨ r))
       • Associativity of ∧
         • ((p ∧ q) ∧ r) ≡ (p ∧ (q ∧ r))
       • Distributivity (∨, ∧)
         • (p ∨ (q ∧ r)) ≡ ((p ∨ q) ∧ (p ∨ r))
       • Neutral element of ∨
         • (p ∨ 0) ≡ p
       • Distributivity (∧, ∨)
         • (p ∧ (q ∨ r)) ≡ ((p ∧ q) ∨ (p ∧ r))
       • 2nd Law of De Morgan
         • [ ∼ (p ∨ q) ] ≡ (∼p ∧ ∼q)
       • Neutral element of ∧
         • (p ∧ 1) ≡ p 
       • Absorption (∧, ∨)
         • (p ∧ (p ∨ q)) ≡ p
       • 1st Law of De Morgan
         • [∼ (p ∧ q) ] ≡ (∼p ∨ ∼q)
       • Absorption (∨, ∧)
         • (p ∨ (p ∧ q)) ≡ p
     • Operations
       • Denial
       • Conjunction
       • Disjunction
       • Biconditional
       • Implication
   • Fuzzy predicates unit
     • 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
     • Examples
       • The sentence "x
         • 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.
       • The statement "x
         • 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.
   • Fuzzy proposition p

6. Fuzzy Sets

   • 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.
   • Finite fuzzy set
     • A = {a1 | uA (a1), a2 | uA (a2), a3 | uA (a3), ..., an | uA (n)}
   • Countably infinite fuzzy set
     • B = {b1|µB(b1), b2|µB(b2), b3|µB(b3), …}
   • Relations between fuzzy sets
     • Equality
       • 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)].
     • Inclusion
       • 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)].
   • Operations with fuzzy sets
     • Complement
       • 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)].
     • Union
       • 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)}.
     • Difference
       • 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 
     • Symmetric difference
       • 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).
     • Intersection
       • 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)}.