Lecture 3. Mathematical Induction
... Induction in natural sciences cannot be absolute, because it is based on a very large but finite number of observations and experiments. We know that in the process of the evolution of such sciences as physics or biology, all the fundamental laws from time to time have been revised. For example, New ...
... Induction in natural sciences cannot be absolute, because it is based on a very large but finite number of observations and experiments. We know that in the process of the evolution of such sciences as physics or biology, all the fundamental laws from time to time have been revised. For example, New ...
What is a Function?
... You can pass zero or several input values You can pass values of different types Each parameter has name Parameters are assigned to particular values when the function is called ...
... You can pass zero or several input values You can pass values of different types Each parameter has name Parameters are assigned to particular values when the function is called ...
Nonmonotonic Logic II: Nonmonotonic Modal Theories
... The first two of these follow by predicate calculus. The third follows because ~CANFLY(FRED) is not a member of the fixed point. In other words, M CANFLY(FRED) is in Astheory(fixed-point). So by the first proper axiom, CANFLY(FRED) is in the fixed point as well. Of course, I have not proven that thi ...
... The first two of these follow by predicate calculus. The third follows because ~CANFLY(FRED) is not a member of the fixed point. In other words, M CANFLY(FRED) is in Astheory(fixed-point). So by the first proper axiom, CANFLY(FRED) is in the fixed point as well. Of course, I have not proven that thi ...
x - peacock
... Ways to Write a Domain/Range Let’s use the function y = 2x, for example, the domain and range of this function is any real number. The domain and range of a function can be expressed using set, interval, and inequality notation. • Set Notation - Domain: {x| x∈ℝ} and Range: {y| y∈ℝ} , where ∈ means ...
... Ways to Write a Domain/Range Let’s use the function y = 2x, for example, the domain and range of this function is any real number. The domain and range of a function can be expressed using set, interval, and inequality notation. • Set Notation - Domain: {x| x∈ℝ} and Range: {y| y∈ℝ} , where ∈ means ...
Using Explicit Formulas for Sequences
... Certificates of Deposites (CDs) that yield a high interest rate paid once a year. Suppose you deposited $28,700 and expected a 4.1% interest rate to be compounded annually. Then the formula Sn = 28,700(1.041)n–1 gives your total savings at any time during the year leading up to the nth anniversary. ...
... Certificates of Deposites (CDs) that yield a high interest rate paid once a year. Suppose you deposited $28,700 and expected a 4.1% interest rate to be compounded annually. Then the formula Sn = 28,700(1.041)n–1 gives your total savings at any time during the year leading up to the nth anniversary. ...
What is a proof? - Computer Science
... Kemp gave a proof that was deemed false 11 years after it was published! His proof, however, contains the essential ideas that were used in subsequent proofs. In our case, we will not learn much from a false proof now, but it will give some insight about the nature of what a proof really is. Conside ...
... Kemp gave a proof that was deemed false 11 years after it was published! His proof, however, contains the essential ideas that were used in subsequent proofs. In our case, we will not learn much from a false proof now, but it will give some insight about the nature of what a proof really is. Conside ...
Lecture_Notes (reformatted)
... visualization of subtle dependencies. This implies that 2SAT is no harder than the Cycles in Graph problem. Note how reductions can be used to show that a problem is easy or hard, depending on the problems to and from which we are reducing. To show a problem is hard, reduce a hard problem to it. To ...
... visualization of subtle dependencies. This implies that 2SAT is no harder than the Cycles in Graph problem. Note how reductions can be used to show that a problem is easy or hard, depending on the problems to and from which we are reducing. To show a problem is hard, reduce a hard problem to it. To ...
Truth-Functional Propositional Logic
... why the rules work or not. The uniformity, simplicity, and regularity of these arithmetical rules, and their applicability with minimal understanding, is shown by the existence of extremely simple artificial devices for effective arithmetical calculation such as the ancient abacus. Before any system ...
... why the rules work or not. The uniformity, simplicity, and regularity of these arithmetical rules, and their applicability with minimal understanding, is shown by the existence of extremely simple artificial devices for effective arithmetical calculation such as the ancient abacus. Before any system ...
How Does Resolution Works in Propositional Calculus and
... A quantifier is a symbol that permits one to declare or identify the range or scope of the variable in a logical expression. There are two basic quantifiers used in logic one is universal quantifier which is denoted by the symbol “” and the other is existential quantifier which is denoted by the sy ...
... A quantifier is a symbol that permits one to declare or identify the range or scope of the variable in a logical expression. There are two basic quantifiers used in logic one is universal quantifier which is denoted by the symbol “” and the other is existential quantifier which is denoted by the sy ...
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: ""The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions.""The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.