CS151 Fall 2014 Lecture 17 – 10/23 Functions
... If you map the set of positive integers to the top row first, then you will not be able to reach the second row. The trick is to visit the rational numbers diagonal by diagonal. Each diagonal is finite, so eventually every pair will be visited. Therefore, there is a bijection from the set of positiv ...
... If you map the set of positive integers to the top row first, then you will not be able to reach the second row. The trick is to visit the rational numbers diagonal by diagonal. Each diagonal is finite, so eventually every pair will be visited. Therefore, there is a bijection from the set of positiv ...
INTRODUCTION TO GROUP THEORY (MATH 10005) The main
... Often, especially when we’re dealing with abstract properties of general groups, we’ll simplify the notation by writing xy instead of x?y, as though we’re multiplying. In this case we’ll say, for example, “Let G be a multiplicatively-written group”. Note that this is purely a matter of the notation ...
... Often, especially when we’re dealing with abstract properties of general groups, we’ll simplify the notation by writing xy instead of x?y, as though we’re multiplying. In this case we’ll say, for example, “Let G be a multiplicatively-written group”. Note that this is purely a matter of the notation ...
On the Infinitude of the Prime Numbers
... • The series 1/1 + 1/2 + 1/4 + .. , -t: 1/2n + ... converges (the sum is 2, as is easily shown). • The series 1/1 + 1/3 + 1/9 + . . . + 1/3n + ... converges (the sum in this case is 3/2). • The series 1+ 1+ 1 + . . , diverges (rather trivially). • The series 1-1 + 1-1 + 1-1 + 1- . .. also fails to c ...
... • The series 1/1 + 1/2 + 1/4 + .. , -t: 1/2n + ... converges (the sum is 2, as is easily shown). • The series 1/1 + 1/3 + 1/9 + . . . + 1/3n + ... converges (the sum in this case is 3/2). • The series 1+ 1+ 1 + . . , diverges (rather trivially). • The series 1-1 + 1-1 + 1-1 + 1- . .. also fails to c ...
Difficulties of the set of natural numbers
... adding more and more numbers can not be exhausted in principle, so it never can make a definite entity. The latter is based one the hypothesis that all natural numbers can form an actual, completed totality, namely, a set. That means the static set has already been completed and contained all natura ...
... adding more and more numbers can not be exhausted in principle, so it never can make a definite entity. The latter is based one the hypothesis that all natural numbers can form an actual, completed totality, namely, a set. That means the static set has already been completed and contained all natura ...
The Probabilistic Method
... Definition. A family F of sets is called intersecting if A, B ∈ F implies A ∩ B 6= ∅, i.e. A, B share a common element. Suppose n ≥ 2k and let F be an intersecting family of k-element subsets of an n-set, for definiteness {0, . . . , n − 1}. ...
... Definition. A family F of sets is called intersecting if A, B ∈ F implies A ∩ B 6= ∅, i.e. A, B share a common element. Suppose n ≥ 2k and let F be an intersecting family of k-element subsets of an n-set, for definiteness {0, . . . , n − 1}. ...
Sample pages 2 PDF
... we can prove are called theorems. Some true statements, however, are so basic that there are no even more basic statements that we can derive them from; these are called axioms. A person who understands decimal addition will clearly be able to answer the following simple Questions: Which of the foll ...
... we can prove are called theorems. Some true statements, however, are so basic that there are no even more basic statements that we can derive them from; these are called axioms. A person who understands decimal addition will clearly be able to answer the following simple Questions: Which of the foll ...
binary digit distribution over naturally defined sequences
... Thus pending proof of the lemma, the theorem is established. ...
... Thus pending proof of the lemma, the theorem is established. ...
PowerPoint file for CSL 02, Edinburgh, UK
... The converse of arrows in the hierarchy of semi-classical principles are conjectured not to be derivable in HA. If the scheme S0n–DNE is not derivable from the scheme P0n–LEM, then the conjecture is proved for the n-level. The conjecture have been solved for n=1, 2 levels, which include all of the L ...
... The converse of arrows in the hierarchy of semi-classical principles are conjectured not to be derivable in HA. If the scheme S0n–DNE is not derivable from the scheme P0n–LEM, then the conjecture is proved for the n-level. The conjecture have been solved for n=1, 2 levels, which include all of the L ...
