Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Predicates. Methods of Proof. They are used in graphs, vector spaces, ring theory, and so on. Introduction to Logic and Set Theory-2013-2014 General Course Notes December 2, 2013 These notes were prepared as an aid to the student. axiomatic set theory with urelements. IV. Why understand set theory and logic applications? Set theory has many applications in mathematics and other fields. 6. Mathematical Logic is a branch of mathematics which is mainly concerned with the relationship between “semantic” concepts (i.e. We will need only a few facts about sets and techniques for dealing with them, which we set out in this section and the next. Predicate Logic and Quantifiers. 0. Mathematical Induction. Logic and Set Theory. Informal Proof. Like logic, the subject of sets is rich and interesting for its own sake. Unique Existence. The intuitive idea of a set is probably even older than that of number. The rules are so simple that … Universal and Existential Quantifiers. Predicates. Indirect Proof. Conditional Proof. III. 4. Proof by Counter Example. In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language.In most scenarios, a deductive system is first understood from context, after which an element ∈ of a theory is then called a theorem of the theory. Expressing infinite elements each equivalence class in First Order logic. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. IV. 1. Imagine that we wanted to represent these … Universal and Existential Quantifiers. The Venn diagram is a good introduction to set theory, because it makes the next part a lot easier to explain. Methods of Proof. III. As opposed to predicate calculus, which will be studied in Chapter 4, the statements will not have quanti er symbols like 8, 9. Almost everyone knows the game of Tic-Tac-Toe, in which players mark X’s and O’s on a three-by-three grid until one player makes three in a row, or the grid gets filled up with no winner (a draw). George Boole. What kind of logic is mine? Informal Proof. These notes were prepared using notes from the course taught by Uri Avraham, Assaf Hasson, and of course, Matti Rubin. Multiple Quantifiers. 2. For example, a deck of cards, every student enrolled in Proof by Counter Example. Multiple Quantifiers. What different possible predicates are there for Peano arithmetic? Unique Existence. Module 6: Set Theory and Logic. Negation of Quantified Predicates. Questions about Peano axioms and second-order logic. Search for: Putting It Together: Set Theory and Logic. Negation of Quantified Predicates. Formal Proof. Set theory, branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions.The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical … Mathematical Induction. Predicate Logic and Quantifiers. Obviously, all programming languages use boolean logic (values are true and false, operators are and, or, not, exclusive or). They are not guaran-teed to be comprehensive of the material covered in the course. 2. 4. Conditional Proof. The subjects of register machines and random access machines have been dropped from Section 5.5 Chapter 5. Defining logic is a bit challenging and it is more like a philosophical endeavor but concisely speaking it is a system rules ( inference rules) that can help us prove and disprove stuff. Indirect Proof. The language of set theory can … Chapter 1 Set Theory 1.1 Basic deﬁnitions and notation A set is a collection of objects. 3. In this module we’ve seen how logic and valid arguments can be formalized using mathematical notation and a few basic rules. Set Theory and Logic Supplementary Materials Math 103: Contemporary Mathematics with Applications A. Calini, E. Jurisich, S. Shields c 2008. mathematical objects) and “syntactic” concepts (such as formal languages, formal deductions and proofs, and computability). Formal Proof. 1 Propositional calculus II Logic and Set Theory 1 Propositional calculus Propositional calculus is the study of logical statements such p)pand p) (q)p). V. Naïve Set Theory. V. Naïve Set Theory. An appendix on second-order logic will give the reader an idea of the advantages and limitations of the systems of first-order logic used in Members of a herd of animals, for example, could be matched with stones in a sack without members Set, In mathematics and logic, any collection of objects (elements), which may be mathematical (e.g., numbers, functions) or not. All these concepts can be defined as sets satisfying specific properties (or axioms) of sets. Axioms of set theory and logic.
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