logic and set theory

1. LOGIC Prepared by: Nathaniel T. Sullano BS Math – 3

2. Logic … is a science that deals with the principles and criteria of validity of inference and demonstration. is the formal systematic study of the principles of valid inference and correct reasoning . Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science .

3. Logic… has became mathematized in the 19 th century, in the work of mostly British mathematicians such as George Peacock (1791-1858), George Boole (1815-1864), William Stanley Jevons (1835-1882), and Augustus de Morgan, and a few Americans, notably Charles Sanders Peirce. was the creation of 19 th -century analysts and geometers, prominent among them is Georg Cantor (1845-1918), whose inspiration came from geometry and analysis, mostly the latter. mathematization of logic has a pre-history that goes back to Leibniz.

4. George Peacock His main contribution to mathematical analysis is his attempt to place algebra on a strictly logical basis . Concluded that the science of algebra has two parts – arithmetical and symbolical algebra. 1791-1858

5. Augustus de Morgan (British/English) developed two laws of negation (disjunction & conjunction) interested, like other mathematicians, in using mathematics to demonstrate logic furthered Boole’s work of incorporating logic and mathematics formally stated the laws of set theory 1806-1871

6. De Morgan’s Laws In formal logic , De Morgan's laws  are rules relating the logical operators "and" and "or" in terms of each other via  negation . With two operands A and B: In another form, NOT (P AND Q ) = (NOT P ) OR (NOT Q) NOT (P OR Q) = (NOT P ) AND (NOT Q ) The rules can be expressed in English as: " The negation of a conjunction is the  disjunction  of the negations. " and " The negation of a disjunction is the conjunction of the negations . "  

7. Symbolic calculus Lagrange’s algebraic approach to analysis (Thinking of Taylor’s Theorem).   Where Df (x) = f’(x), and comparing with the Taylor’s series of the exponential function,   Lagrange arrived at the formal equation   Converse relation,  

8. George Boole self‑taught mathematician with an interest in logic developed an algebra of logic (Boolean Algebra) featured the operators and or not nor (exclusive or) 1815-1864

9. Boole’s Mathematical Analysis of Logic Boole soon began to see the possibilities for applying his algebra to the solution of logical problems. Boole's 1847 work, 'The Mathematical Analysis of Logic', not only expanded on Gottfried Leibniz' earlier speculations on the correlation between logic and math, but argued that logic was principally a discipline of mathematics, rather than philosophy. He wrote that, “the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed but solely upon the laws of their combination…”

10. Logic and Classes Boole denoted a generic member of a class by an uppercase ‘ X’ , and used the lowercase ‘ x’. Then “ xy ” was to denote the class “whose members are both X’s and Y’s” This language rather blurs the distinction between a set, its members, and the properties that determine what the members are.

11. Boole’s Laws of Thought The Laws of Thought began with a very general proposition that laid out the universe of symbols to be used. These are: 1 st . Literal symbols, as x, y, & z, representing things as subjects of our conceptions. 2 nd . Signs of operation, as , standing for those operations of the mind by which the conceptions of things are combined or resolved so as to form new conceptions involving the same elements. 3 rd . The sign of identity, =.  

12. … Laws of Thought Boole used + to represent disjunction (or) and juxtaposition, used in algebra for multiplication, to represent conjunction (and). The sign was used to stand for “and not”. In his examples, he used + only when the properties were disjoint and only when the property subtracted was a subset of the property from which it was subtracted. Example: “the equivalence of European men and women with European men and European women ” as the equation “the class of men who are non-Asiatic and white is the same as the class of white men who are not white Asiatic men.”  

13. Set Theory

14. …Set Theory It is the common language used to express concepts in all areas of mathematics. It is also useful in analyzing difficult concepts in mathematics and logic.

15. Why Study Set Theory? Understanding set theory helps people to … see things in terms of systems organize things into groups begin to understand logic

16. Key Mathematicians These mathematicians influenced the development of set theory and logic: Georg Cantor John Venn George Boole Augustus De Morgan

17. Georg Cantor 1845 -1918 Founder and developed set theory set theory was not initially accepted because it was radically different set theory today is widely accepted and is used in many areas of mathematics

18. …Cantor the concept of infinity was expanded by Cantor’s set theory Cantor proved there are “levels of infinity” an infinitude of integers initially ending with  or an infinitude of real numbers exist between 1 and 2; there are more real numbers than there are integers …  

19. John Venn 1834-1923 studied and taught logic and probability theory articulated Boole’s algebra of logic devised a simple way to diagram set operations (Venn Diagrams )

20. Technical background        

21. Basic Set Theory Definitions A set is a collection of elements An element is an object contained in a set If every element of Set A is also contained in Set B , then Set A is a subset of Set B A is a proper subset of B if B has more elements than A does The universal set contains all of the elements relevant to a given discussion

22. Set Theory Notation Symbol Meaning Upper case designates set name Lower case designates set elements { } enclose elements in set  or is (or is not) an element of  is a subset of ( includes equal sets)  is a proper subset of  is not a subset of  is a superset of | or : such that (if a condition is true) | | the cardinality of a set

23. Sets (or Group) sets can be defined in two ways: by listing each element (Roster Method) by defining the rules for membership (Rule Method) Examples: A = {2,4,6,8,10} A = { x|x is a positive even integer <12 }

24. Ordinal Numbers (or just Ordinals ) Numbers that indicates the order or position of something in a group or set. were introduced by Georg Cantor  to accommodate infinite sequences and to classify sets with certain kinds of order structures on them .

25. Cardinal Numbers (or just Cardinals ) Is a number used in simple counting and that indicates how many elements there are in a set or assemblage. Cardinality refers to the number of elements in a set.

26. Cantor- Bendixson Theorem Is a major theorem of set theory. states that closed sets of a Polish space  X  have the perfect set property in a particularly strong form; any closed set  C  may be written uniquely as the disjoint union of a perfect set  P  and a countable set  S . Georg Cantor and   Ivar Otto  Bendixson