BASIC SET THEORY
... NOTE: An object is either in a set or not in it. Nothing can be "half in" , "in twice", etc. {A,B,C} and {A,A,B,C} are the same set. If set A and set B have the same elements, then by definition they are two names for the SAME SET. If either has an element the other doesn't, they are DIFFERENT SETS ...
... NOTE: An object is either in a set or not in it. Nothing can be "half in" , "in twice", etc. {A,B,C} and {A,A,B,C} are the same set. If set A and set B have the same elements, then by definition they are two names for the SAME SET. If either has an element the other doesn't, they are DIFFERENT SETS ...
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... The first approach is axiomatic and abstract. We state logical properties of the numbers using first-order logic. It might be the case that these first-order properties describe numbers so well that they capture our intuition completely. The classical first-order theory of numbers is called Peano Ar ...
... The first approach is axiomatic and abstract. We state logical properties of the numbers using first-order logic. It might be the case that these first-order properties describe numbers so well that they capture our intuition completely. The classical first-order theory of numbers is called Peano Ar ...
Discrete Math 6A
... First loop over y and and for every y loop over x. For every value of y, check if P(x,y) is true for all x. If you found one, the proposition must be true. ...
... First loop over y and and for every y loop over x. For every value of y, check if P(x,y) is true for all x. If you found one, the proposition must be true. ...
Document
... The significance of Russell's paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. For example, assuming both P and ~P, any arbitrary proposition, Q, can be proved as follows: from P we obtain P Q by the rule of Addition; then from P Q and ~ ...
... The significance of Russell's paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. For example, assuming both P and ~P, any arbitrary proposition, Q, can be proved as follows: from P we obtain P Q by the rule of Addition; then from P Q and ~ ...
PRESERVATION THEOREMS IN LUKASIEWICZ MODEL THEORY
... When we work on BL-algebras with domain [0, 1], the operation ∗ is called a tnorm. In this case the BL-algebra is called standard or t-norm BL-algebra, see [7] and [8]. In the sequel, we work with a special but important BL-algebra, i.e. Lukasiewicz BL-algebra which is equipped with the Lukasiewicz ...
... When we work on BL-algebras with domain [0, 1], the operation ∗ is called a tnorm. In this case the BL-algebra is called standard or t-norm BL-algebra, see [7] and [8]. In the sequel, we work with a special but important BL-algebra, i.e. Lukasiewicz BL-algebra which is equipped with the Lukasiewicz ...
Stephen Cook and Phuong Nguyen. Logical foundations of proof
... proofs in P. And a sort of converse to this last statement holds too since the theory T proves the soundness of P. Thus, in the language of the previous paragraph, the proof system P is not only complete, but efficiently so, with respect to the propositional translations of bounded theorems in T . A ...
... proofs in P. And a sort of converse to this last statement holds too since the theory T proves the soundness of P. Thus, in the language of the previous paragraph, the proof system P is not only complete, but efficiently so, with respect to the propositional translations of bounded theorems in T . A ...
Sets
... Proper subset: A ⊂ B – If A is a subset of B and there is at least one element of B that is not contained in A, the A is a proper subset of B. How to find the number of subsets: If a set contains n elements, then if contains 2n subsets. How to find the number of proper subsets: If a set contains n e ...
... Proper subset: A ⊂ B – If A is a subset of B and there is at least one element of B that is not contained in A, the A is a proper subset of B. How to find the number of subsets: If a set contains n elements, then if contains 2n subsets. How to find the number of proper subsets: If a set contains n e ...
Proposition: The following properties hold A ∩ B ⊆ A, A ∩ B ⊆ B, A
... Definition: If A is a finite set we write |A| for the number of elements in A and call |A| the cardinality of A. Proposition: If A and B are finite sets then |A ∪ B| = |A| + |B| − |A ∩ B| Proof: From a Venn diagram we see |A ∪ B| = |A∩ ∼B| + |A ∩ B| + | ∼A ∩ B| The first two terms give |A| while the ...
... Definition: If A is a finite set we write |A| for the number of elements in A and call |A| the cardinality of A. Proposition: If A and B are finite sets then |A ∪ B| = |A| + |B| − |A ∩ B| Proof: From a Venn diagram we see |A ∪ B| = |A∩ ∼B| + |A ∩ B| + | ∼A ∩ B| The first two terms give |A| while the ...
3-8 Unions and Intersection of Sets
... In your left pocket, you have a quarter, a paper clip, and a key. In your right pocket, you have a penny, a quarter, a pencil, and a marble. What is a set that represents the different items in your pockets ...
... In your left pocket, you have a quarter, a paper clip, and a key. In your right pocket, you have a penny, a quarter, a pencil, and a marble. What is a set that represents the different items in your pockets ...
Number systems. - Elad Aigner
... and π. Often these are defined as those numbers whose decimal expansion is non-periodic. The irrational arise naturally. ...
... and π. Often these are defined as those numbers whose decimal expansion is non-periodic. The irrational arise naturally. ...
On the regular extension axiom and its variants
... The first interesting consequence of wREA is that the class of hereditarily countable sets, HC = H(ω ∪ {ω}), constitutes a set. In the Leeds-Manchester Proof Theory Seminar, Peter Aczel asked whether CZF is at least strong enough to show that HC is a set. This section is devoted to showing that this ...
... The first interesting consequence of wREA is that the class of hereditarily countable sets, HC = H(ω ∪ {ω}), constitutes a set. In the Leeds-Manchester Proof Theory Seminar, Peter Aczel asked whether CZF is at least strong enough to show that HC is a set. This section is devoted to showing that this ...
MATH 251
... discussed include logic of compound and quantified statements, number theory and methods of proof, mathematical induction, counting methods, finite state automata, functions and relations. 2. Course Objectives (Overall Aims of Course): Upon completion of this course, students should be able to: 1. U ...
... discussed include logic of compound and quantified statements, number theory and methods of proof, mathematical induction, counting methods, finite state automata, functions and relations. 2. Course Objectives (Overall Aims of Course): Upon completion of this course, students should be able to: 1. U ...
Set theory
Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.