p - Erwin Sitompul
... A formal proof is a set of proofs which follows logically from the set of premises. Formal proofs allow us to infer new true statements from known true statements. A proposition or its part can be transformed using a sequence of logical equivalence until some conclusions can be reached. Exam ...
... A formal proof is a set of proofs which follows logically from the set of premises. Formal proofs allow us to infer new true statements from known true statements. A proposition or its part can be transformed using a sequence of logical equivalence until some conclusions can be reached. Exam ...
PPT for Section 2.8, 2.9
... If a < b is true, then ac < bc is also true. However, if c < 0, then a bc The above principle can be remembered as “multiplying an inequality by a negative number results in the inequality sign being reversed.” Example: a = 2, b = 3, c = 4. Then, a < b is true because 2 ...
... If a < b is true, then ac < bc is also true. However, if c < 0, then a bc The above principle can be remembered as “multiplying an inequality by a negative number results in the inequality sign being reversed.” Example: a = 2, b = 3, c = 4. Then, a < b is true because 2 ...
countably infinite
... aleph null = ﭏ0. If a set is not countable we say it is uncountable. P. 1 ...
... aleph null = ﭏ0. If a set is not countable we say it is uncountable. P. 1 ...
CSC 111 Exam I. Spring 2008. NAME: March 11, 2008 Open Book
... c) What do the following Python statements produce? x = "Buckaroo Bonzai" x = x[5:7] + x[5:7] print x i) 'arar' ii) 'roro' iii) 'rooroo' iv) None of the above. d) What does the following definite loop produce? for i in [19,11,5]: print i - 10, i) 9 1 -5 ii) 9 ...
... c) What do the following Python statements produce? x = "Buckaroo Bonzai" x = x[5:7] + x[5:7] print x i) 'arar' ii) 'roro' iii) 'rooroo' iv) None of the above. d) What does the following definite loop produce? for i in [19,11,5]: print i - 10, i) 9 1 -5 ii) 9 ...
Section 4.7 Inverse Trig Functions
... Keep in mind that inverse functions can only exist if the original function is one-to-one. Are the arcsine or cosine functions one-to-one? ...
... Keep in mind that inverse functions can only exist if the original function is one-to-one. Are the arcsine or cosine functions one-to-one? ...
On Provability Logic
... proved above. So PA 6` ¬ν. Proposition 1 is basically Gödel first incompleteness theorem, while Proposition 2 below is Gödel second incompleteness theorem. Proposition 2 (a) If PA ` ν ≡ ¬Pr(ν), then PA ` Con ≡ ν. So PA 6` Con. (b) PA ` Con → ¬Pr(Con). We are not going to prove Proposition 2. We on ...
... proved above. So PA 6` ¬ν. Proposition 1 is basically Gödel first incompleteness theorem, while Proposition 2 below is Gödel second incompleteness theorem. Proposition 2 (a) If PA ` ν ≡ ¬Pr(ν), then PA ` Con ≡ ν. So PA 6` Con. (b) PA ` Con → ¬Pr(Con). We are not going to prove Proposition 2. We on ...
MEASUREMENT
... Scientific Notation Science often involves working with very large or very small numbers. Ex. Speed of light = 300,000,000 m/s Speed of snail = 0.00086 m/s Scientific notation makes very large or very small numbers easier to work with. ...
... Scientific Notation Science often involves working with very large or very small numbers. Ex. Speed of light = 300,000,000 m/s Speed of snail = 0.00086 m/s Scientific notation makes very large or very small numbers easier to work with. ...
MATH 1113 Review Sheet for the Final Exam
... Inverse Function of a function; connection between domains and ranges of these functions Defining the terms one to one and one to one function o Intuitively, one to one means no partner sharing o Determining when a graph that represents a function is one to one; the horizontal line test Relati ...
... Inverse Function of a function; connection between domains and ranges of these functions Defining the terms one to one and one to one function o Intuitively, one to one means no partner sharing o Determining when a graph that represents a function is one to one; the horizontal line test Relati ...
MATH 1113 Review Sheet for the Final Exam
... Inverse Function of a function; connection between domains and ranges of these functions Defining the terms one to one and one to one function o Intuitively, one to one means no partner sharing o Determining when a graph that represents a function is one to one; the horizontal line test Relati ...
... Inverse Function of a function; connection between domains and ranges of these functions Defining the terms one to one and one to one function o Intuitively, one to one means no partner sharing o Determining when a graph that represents a function is one to one; the horizontal line test Relati ...
MATH 311–01 Exam #1 Solutions 1. (7 points) Consider the true
... This is impossible. If B is a subset of N, then all of B’s elements are natural numbers, but the requirement that A ∈ B would necessitate that at least one element of B would be a set. (c) Finite sets X, Y , and Z such that X ⊆ Z, Y ⊆ Z, |X| = |Y |, and |X ∩ Y | = 2. The requirement here is to simpl ...
... This is impossible. If B is a subset of N, then all of B’s elements are natural numbers, but the requirement that A ∈ B would necessitate that at least one element of B would be a set. (c) Finite sets X, Y , and Z such that X ⊆ Z, Y ⊆ Z, |X| = |Y |, and |X ∩ Y | = 2. The requirement here is to simpl ...
Haskell 5A
... The early functional language Lisp took the approach of dynamic scoping, where non-local variables refer to the closest definition of that variable at the point where the function is executed (we’ll do an exercise later). Proper support for lexically scoped first-class functions was introduced in Sc ...
... The early functional language Lisp took the approach of dynamic scoping, where non-local variables refer to the closest definition of that variable at the point where the function is executed (we’ll do an exercise later). Proper support for lexically scoped first-class functions was introduced in Sc ...
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.