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Este Cmap, tiene información relacionada con: RAMAS DE LA LOGICA, THE RELATIONSHIP BETWEEN THE THREE BRANCHES ???? MAIN RULES OF INFERENCE MPP Modus ponendoponens A → B A - - - - - B MTTModustollendotollens A → B ¬B - - - - - ¬A SD Silogismo Disyuntivo A ∨ B ¬A - - - - - ¬B SH Silogismo hipotético A → B B → C - - - - - A → C LS Ley de simplificación A ∧ B - - - - - A LA Ley de adición A - - - - - A ∨ B CONTRAPOSITIVA A → B - - - - - ¬B → ¬A The verification of the previous rules is direct and it is enough to make a formula with the conjunction of the premises conditional the conclusion and prove that it is a tautology, for example making a table and obtaining all the true values., TYPES OF INFERENCE Inductive Deductive Transductive Abductive ???? In logic, especially in mathematical logic, an inference rule is a scheme for constructing valid inferences. These schemes establish syntactic relationships between a set of formulas called premises and an assertion called conclusion., AXIOMS Axioms 1. There is a set G of objects, subject to an equivalence relation, denoted by "=" that satisfies the substitution principle. This means that if a = b, b can replace a in any expression that contains it, without altering the validity of the expression. 2. (a) A combination rule "+" is defined in such a way that a + b is in G whenever at least a or b are. (b) A combination rule "." is defined in such a way that a . b is in G whenever both a and b are. 3. Neutrals (a) There is an element 0 in G such that for each a of G: a + 0 = a (b) There is an element 1 in G such that for each a of G: a . 1 = a 4. Commutatives. For all pairs of elements a and b belonging to G, the following is true: (a) a + b = b + a (b) a . b = b . a 5. Distributive. For all three of elements a, b, c belonging to G, a (a) a + (b . c) = (a + b) . (a + c) b (b) a . (b + c) is fulfilled = a. b + a . c 6. Complement. For each element a of G there is an element a such that: a. a= o a+a=1 7. There are at least two elements x, and in G such that x <> and There is similarity of many of these postulates with those of common algebra. However, the first of the distributive rules (on the sum) and the existence of the complement fundamentally differentiate this algebra from the common one. ???? BOOLEAN ALGEBRA Boolean algebra is a tool of fundamental importance in the world of computing. The properties that are verified in it serve as the basis for the design and construction of computers that work with objects whose values are discrete , that is, digital computers, particularly binary ones (in which basic objects have only 2 possible values) which are, in short, the totality of computers in current use. Like all algebra, Boole's part of an axiomatic body, which can take different forms, varying the quantity and quality of the axioms. Here in particular we will take one: the one proposed by Huntington in 1904 that has the advantage of being consistent and independent., PROPERTIES Duality If we analyze the postulates we will see that they are presented in pairs and in such a way that one of the couple is obtained from another by changing "0" to "1" together with "+" to "." (and vice versa). This ensures that each property that is demonstrated in this Algebra has a "dual" that is also true (to demonstrate the dual it would be enough to repeat the demonstration done substituting each postulate or property used by its dual). ASSOCIATIVE Although the associative laws are often included within the axiomatic body, in fact they are demonstrable from the axioms presented here (demonstration that we will not do) for which we present them as properties. A) a + (b + c) = (a + b) + c B) a. (b.c) = (a.b) .c IDEMPOTENCE is the property to perform a certain action several times and still get the same result that would be obtained if it were performed only once a + a = a a.a = a ???? THE RELATIONSHIP BETWEEN THE THREE BRANCHES, INFERENCE Inference is the way in which we obtain conclusions based on data and statements established ???? RELATIONSHIP OF THE MATHEMATICAL LOGIC SYSTEM, AXIOMS Axioms 1. There is a set G of objects, subject to an equivalence relation, denoted by "=" that satisfies the substitution principle. This means that if a = b, b can replace a in any expression that contains it, without altering the validity of the expression. 2. (a) A combination rule "+" is defined in such a way that a + b is in G whenever at least a or b are. (b) A combination rule "." is defined in such a way that a . b is in G whenever both a and b are. 3. Neutrals (a) There is an element 0 in G such that for each a of G: a + 0 = a (b) There is an element 1 in G such that for each a of G: a . 1 = a 4. Commutatives. For all pairs of elements a and b belonging to G, the following is true: (a) a + b = b + a (b) a . b = b . a 5. Distributive. For all three of elements a, b, c belonging to G, a (a) a + (b . c) = (a + b) . (a + c) b (b) a . (b + c) is fulfilled = a. b + a . c 6. Complement. For each element a of G there is an element a such that: a. a= o a+a=1 7. There are at least two elements x, and in G such that x <> and There is similarity of many of these postulates with those of common algebra. However, the first of the distributive rules (on the sum) and the existence of the complement fundamentally differentiate this algebra from the common one. ???? BOOLEAN ALGEBRA Boolean algebra is a tool of fundamental importance in the world of computing. The properties that are verified in it serve as the basis for the design and construction of computers that work with objects whose values are discrete , that is, digital computers, particularly binary ones (in which basic objects have only 2 possible values) which are, in short, the totality of computers in current use. Like all algebra, Boole's part of an axiomatic body, which can take different forms, varying the quantity and quality of the axioms. Here in particular we will take one: the one proposed by Huntington in 1904 that has the advantage of being consistent and independent., PROPERTIES Duality If we analyze the postulates we will see that they are presented in pairs and in such a way that one of the couple is obtained from another by changing "0" to "1" together with "+" to "." (and vice versa). This ensures that each property that is demonstrated in this Algebra has a "dual" that is also true (to demonstrate the dual it would be enough to repeat the demonstration done substituting each postulate or property used by its dual). ASSOCIATIVE Although the associative laws are often included within the axiomatic body, in fact they are demonstrable from the axioms presented here (demonstration that we will not do) for which we present them as properties. A) a + (b + c) = (a + b) + c B) a. (b.c) = (a.b) .c IDEMPOTENCE is the property to perform a certain action several times and still get the same result that would be obtained if it were performed only once a + a = a a.a = a ???? ARITHMETIC MODEL The simplest example of Boolean algebra is composed of a set G of 2 elements: "0" and "1". Naturally, these two elements must coincide with the neutrals of the combination rules to satisfy axiom 3. The combination rules must be defined in order to satisfy the axioms. According to Axiom 4 0+1=1 0.1= 0 And taking into account axiom 5 1+ (1.0)= (1+1). (1+0) (Por axioma 3) (5ª Con a=1, b=1, c=0) 1 +0 = (1+1). 1 0. (0+1)=0.0+0.1 (por axioma 3) (5b con a=0 b=0 c=1) 0.1=0.0+1 Por lo tanto las reglas asociativas son: 0+0=0 0.0=0 0+1=1 0.1=0 1+0=1 1.0=0 1+1=1 1.1=1, TRANSDUCTIVE (from particular to particular or from general to general) It is similar to deductive, it also uses the strategy of analyzing all the possibilities, but in this case there are several cases that can be presented, ???? TYPES OF INFERENCE Inductive Deductive Transductive Abductive, Between mathematical logic and set theory they share logical laws both for sets and for propositional logic. Boolean algebra was an attempt to use algebraic techniques to deal with expressions of propositional logic. At present, Boolean algebra is applied in a generalized way in the field of electronic design. ???? THE RELATIONSHIP BETWEEN THE THREE BRANCHES, DEDUCTIVE (from the general to the particular): In this case there are MPP: Modus PonendoPonens and MTT: Modus TollendoTollens that according to the truth table of the conditional are two ways to establish a valid inference ???? TYPES OF INFERENCE Inductive Deductive Transductive Abductive, INDUCTIVE Inductive inference is the general law that is obtained from the observation of one or more cases and it can not be assured that the conclusion is true in general ???? TYPES OF INFERENCE Inductive Deductive Transductive Abductive, MAIN RULES OF INFERENCE MPP Modus ponendoponens A → B A - - - - - B MTTModustollendotollens A → B ¬B - - - - - ¬A SD Silogismo Disyuntivo A ∨ B ¬A - - - - - ¬B SH Silogismo hipotético A → B B → C - - - - - A → C LS Ley de simplificación A ∧ B - - - - - A LA Ley de adición A - - - - - A ∨ B CONTRAPOSITIVA A → B - - - - - ¬B → ¬A The verification of the previous rules is direct and it is enough to make a formula with the conjunction of the premises conditional the conclusion and prove that it is a tautology, for example making a table and obtaining all the true values. ???? DEDUCTIVE (from the general to the particular): In this case there are MPP: Modus PonendoPonens and MTT: Modus TollendoTollens that according to the truth table of the conditional are two ways to establish a valid inference, ARITHMETIC MODEL The simplest example of Boolean algebra is composed of a set G of 2 elements: "0" and "1". Naturally, these two elements must coincide with the neutrals of the combination rules to satisfy axiom 3. The combination rules must be defined in order to satisfy the axioms. According to Axiom 4 0+1=1 0.1= 0 And taking into account axiom 5 1+ (1.0)= (1+1). (1+0) (Por axioma 3) (5ª Con a=1, b=1, c=0) 1 +0 = (1+1). 1 0. (0+1)=0.0+0.1 (por axioma 3) (5b con a=0 b=0 c=1) 0.1=0.0+1 Por lo tanto las reglas asociativas son: 0+0=0 0.0=0 0+1=1 0.1=0 1+0=1 1.0=0 1+1=1 1.1=1 ???? AXIOMS Axioms 1. There is a set G of objects, subject to an equivalence relation, denoted by "=" that satisfies the substitution principle. This means that if a = b, b can replace a in any expression that contains it, without altering the validity of the expression. 2. (a) A combination rule "+" is defined in such a way that a + b is in G whenever at least a or b are. (b) A combination rule "." is defined in such a way that a . b is in G whenever both a and b are. 3. Neutrals (a) There is an element 0 in G such that for each a of G: a + 0 = a (b) There is an element 1 in G such that for each a of G: a . 1 = a 4. Commutatives. For all pairs of elements a and b belonging to G, the following is true: (a) a + b = b + a (b) a . b = b . a 5. Distributive. For all three of elements a, b, c belonging to G, a (a) a + (b . c) = (a + b) . (a + c) b (b) a . (b + c) is fulfilled = a. b + a . c 6. Complement. For each element a of G there is an element a such that: a. a= o a+a=1 7. There are at least two elements x, and in G such that x <> and There is similarity of many of these postulates with those of common algebra. However, the first of the distributive rules (on the sum) and the existence of the complement fundamentally differentiate this algebra from the common one., THEORY OF SETS A set is a collection of elements with similar characteristics considered in itself as an object. The elements of a set can be the following: people, numbers, colors, letters, figures, etc. It is said that an element (or member) belongs to the set if it is defined as included in some way within it. Example: the set of colors of the rainbow is: AI = {Red, Orange, Yellow, Green, Blue, Indigo, Violet} ???? RELATIONSHIP OF THE MATHEMATICAL LOGIC SYSTEM, BOOLEAN ALGEBRA Boolean algebra is a tool of fundamental importance in the world of computing. The properties that are verified in it serve as the basis for the design and construction of computers that work with objects whose values are discrete , that is, digital computers, particularly binary ones (in which basic objects have only 2 possible values) which are, in short, the totality of computers in current use. Like all algebra, Boole's part of an axiomatic body, which can take different forms, varying the quantity and quality of the axioms. Here in particular we will take one: the one proposed by Huntington in 1904 that has the advantage of being consistent and independent. ???? RELATIONSHIP OF THE MATHEMATICAL LOGIC SYSTEM, PROPERTIES In standard axiomatic set theory, by the Axiom of intentionality, two sets are equal if they have the same elements; therefore there can only be one set without any element. Therefore, there is only a single empty set, and we speak of "the empty set" instead of "an empty set". For any set A:V ???? THE RELATIONSHIP BETWEEN THE THREE BRANCHES, PROPERTIES In standard axiomatic set theory, by the Axiom of intentionality, two sets are equal if they have the same elements; therefore there can only be one set without any element. Therefore, there is only a single empty set, and we speak of "the empty set" instead of "an empty set". For any set A:V ???? THEORY OF SETS A set is a collection of elements with similar characteristics considered in itself as an object. The elements of a set can be the following: people, numbers, colors, letters, figures, etc. It is said that an element (or member) belongs to the set if it is defined as included in some way within it. Example: the set of colors of the rainbow is: AI = {Red, Orange, Yellow, Green, Blue, Indigo, Violet}, In logic, especially in mathematical logic, an inference rule is a scheme for constructing valid inferences. These schemes establish syntactic relationships between a set of formulas called premises and an assertion called conclusion. ???? INFERENCE Inference is the way in which we obtain conclusions based on data and statements established