INTRODUCTION TO GROUP THEORY (MATH 10005) The main
... There are many ways to make precise what we mean by a symmetry. For example, considering the polygon as a subset X of R2 , we could look at bijections F : X → X that preserve distance: i.e., such that the distance between F(x) and F(y) is the same as the distance between x and y. It’s not hard to se ...
... There are many ways to make precise what we mean by a symmetry. For example, considering the polygon as a subset X of R2 , we could look at bijections F : X → X that preserve distance: i.e., such that the distance between F(x) and F(y) is the same as the distance between x and y. It’s not hard to se ...
exam review
... 7. On the island of knights and knaves, you meet three natives, A, B, and C, who address you as follows: A: At least one of us is a knave. B: At most two of us are knaves. What are A, B, and C? Justify your answer (e.g., by using a truth table). ...
... 7. On the island of knights and knaves, you meet three natives, A, B, and C, who address you as follows: A: At least one of us is a knave. B: At most two of us are knaves. What are A, B, and C? Justify your answer (e.g., by using a truth table). ...
math 55: homework #2 solutions - Harvard Mathematics Department
... already in the process of analyzing the first case, so we will turn our attention to the second. In this case, the string preceding 111111111 must have consisted entirely of alternating digits. In particular, there must have been some digit i which was a 1 — we will write this as our first digit, wh ...
... already in the process of analyzing the first case, so we will turn our attention to the second. In this case, the string preceding 111111111 must have consisted entirely of alternating digits. In particular, there must have been some digit i which was a 1 — we will write this as our first digit, wh ...
Principle of Mathematical Induction
... 1. “The Towers of Hanoi” is a puzzle with 3 nails and 7 rings, all of different sizes. Initially all rings are on the same nail in decreasing order from the bottom to the top. The procedure is removing the top ring from any nail and placing it on another nail. It is not allowed to place a bigger rin ...
... 1. “The Towers of Hanoi” is a puzzle with 3 nails and 7 rings, all of different sizes. Initially all rings are on the same nail in decreasing order from the bottom to the top. The procedure is removing the top ring from any nail and placing it on another nail. It is not allowed to place a bigger rin ...
210ch2 - Dr. Djamel Bouchaffra
... Note: f associates with each x in A one and only one y in B. A is called the domain and B is called the codomain. If f(x) = y y is called the image of x under f x is called a preimage of y (note there may be more than one preimage of y but there is only one image of x). The range of f is the set of ...
... Note: f associates with each x in A one and only one y in B. A is called the domain and B is called the codomain. If f(x) = y y is called the image of x under f x is called a preimage of y (note there may be more than one preimage of y but there is only one image of x). The range of f is the set of ...
Section 2.2: Introduction to the Logic of Quantified Statements
... This seems like a silly statement to make, but we need predicate logic to be consistent with propositional logic. Moreover, when we consider examples, it is more clear why it seems reasonable. Example 2.6. Consider the statement “∀ integers x, if x is odd and even, then x is made of cheese.” Clearly ...
... This seems like a silly statement to make, but we need predicate logic to be consistent with propositional logic. Moreover, when we consider examples, it is more clear why it seems reasonable. Example 2.6. Consider the statement “∀ integers x, if x is odd and even, then x is made of cheese.” Clearly ...
The substitutional theory of logical consequence
... to the nonlogical expressions plus the specification of a domain. There is another important difference: The model-theoretic analysis of validity relies on a set-theoretic definition of truth in a model. The substitutional account, in contrast, requires an ‘absolute’ notion of truth that is not rela ...
... to the nonlogical expressions plus the specification of a domain. There is another important difference: The model-theoretic analysis of validity relies on a set-theoretic definition of truth in a model. The substitutional account, in contrast, requires an ‘absolute’ notion of truth that is not rela ...
Problems
... 1. All M are P, some S are not M then: Some S are P. 2. All M are P, some S are not M then: Some S are not P 3. All P are M, some S are not M then: Some S are not P. 2.7.3. Problem 3 (Introduction) In a meeting there are 100 politicians discussing with each other. Everyone of them is either corrupte ...
... 1. All M are P, some S are not M then: Some S are P. 2. All M are P, some S are not M then: Some S are not P 3. All P are M, some S are not M then: Some S are not P. 2.7.3. Problem 3 (Introduction) In a meeting there are 100 politicians discussing with each other. Everyone of them is either corrupte ...
A Conditional Logical Framework *
... could be hardly captured by a rigid type discipline, where bad terms and hypotheses are ruled out a priori, see e.g. [NPP08]. In this paper we develop all the metatheory of LFK . In particular, we prove subject reduction, strong normalization, confluence; this latter under the sole assumption that t ...
... could be hardly captured by a rigid type discipline, where bad terms and hypotheses are ruled out a priori, see e.g. [NPP08]. In this paper we develop all the metatheory of LFK . In particular, we prove subject reduction, strong normalization, confluence; this latter under the sole assumption that t ...
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.