Sets and Functions
... Logarithmic function log x The function sqrt (x) Question. Which one grows faster? Log x or sqrt (x)? Learn about these from the book (and from other sources). ...
... Logarithmic function log x The function sqrt (x) Question. Which one grows faster? Log x or sqrt (x)? Learn about these from the book (and from other sources). ...
On sum-sets and product-sets of complex numbers
... A similar argument works for quaternions and for other hypercomplex numbers. In general, if T and Q are sets of similarity transformations and A is a set of points in space such that from any quadruple (t(p1 ), t(p2 ), q(p1 ), q(p2 )) the elements t ∈ T , q ∈ Q, and p1 6= p2 ∈ A are uniquely determi ...
... A similar argument works for quaternions and for other hypercomplex numbers. In general, if T and Q are sets of similarity transformations and A is a set of points in space such that from any quadruple (t(p1 ), t(p2 ), q(p1 ), q(p2 )) the elements t ∈ T , q ∈ Q, and p1 6= p2 ∈ A are uniquely determi ...
2010 U OF I MOCK PUTNAM EXAM Solutions
... modulo d unchanged, so all points Pi on the path must have x and y-coordinates congruent to the x and y-coordinates of the starting point P0 . Since d ≥ 2, these points do not represent all lattice points, so the path is not a lattice traversal in the sense of the problem. If d = 10, then the possib ...
... modulo d unchanged, so all points Pi on the path must have x and y-coordinates congruent to the x and y-coordinates of the starting point P0 . Since d ≥ 2, these points do not represent all lattice points, so the path is not a lattice traversal in the sense of the problem. If d = 10, then the possib ...
sets and elements
... A set could have as many entries as you would like. It could have one entry, 10 entries, 15 entries, infinite number of entries, or even have no entries at all! For example, in the above list the English alphabet would have 26 entries, while the set of even numbers would have an infinite number of e ...
... A set could have as many entries as you would like. It could have one entry, 10 entries, 15 entries, infinite number of entries, or even have no entries at all! For example, in the above list the English alphabet would have 26 entries, while the set of even numbers would have an infinite number of e ...
Finite MTL
... A totally ordered MTL-algebra (MTL-chain) is archimedean if for every x ≤ y < 1, there exists n ∈ N such that y n ≤ x. A forest is a poset X such that for every a ∈ X the set ↓ a = {x ∈ X | x ≤ a} is a totally ordered subset of X. A p-morphism is a morphism of posets f : X → Y satisfying the followi ...
... A totally ordered MTL-algebra (MTL-chain) is archimedean if for every x ≤ y < 1, there exists n ∈ N such that y n ≤ x. A forest is a poset X such that for every a ∈ X the set ↓ a = {x ∈ X | x ≤ a} is a totally ordered subset of X. A p-morphism is a morphism of posets f : X → Y satisfying the followi ...
First Class - shilepsky.net
... Section 4: Groups Recall when we studied relations we discussed different properties relation could have. For example, one might be reflexive or perhaps it is a function. We discovered that three properties reflexive, symmetric and transitive - combined to make a very important and useful relation, ...
... Section 4: Groups Recall when we studied relations we discussed different properties relation could have. For example, one might be reflexive or perhaps it is a function. We discovered that three properties reflexive, symmetric and transitive - combined to make a very important and useful relation, ...
simplifying expressions
... DISTRIBUTIVE PROPERTY – a way to simplify an expression that is multiplying a single term by a group of terms NOTES and EXAMPLES The distributive property is a very important property in mathematics, which allows us to simplify expressions in the following manner: ***For any numbers a, b and c… ...
... DISTRIBUTIVE PROPERTY – a way to simplify an expression that is multiplying a single term by a group of terms NOTES and EXAMPLES The distributive property is a very important property in mathematics, which allows us to simplify expressions in the following manner: ***For any numbers a, b and c… ...
Putnam Questions, Week 2 1. Prove that the number of subsets of {1
... 2. In how many ways can two squares be selected from an 8-by-8 chessboard so that they are not in the same row or the same column? 3. In how many ways can four squares, not all in the same row or column, be selected from an 8-by-8 chessboard to form a rectangle? 4. Find the number of subsets of {1, ...
... 2. In how many ways can two squares be selected from an 8-by-8 chessboard so that they are not in the same row or the same column? 3. In how many ways can four squares, not all in the same row or column, be selected from an 8-by-8 chessboard to form a rectangle? 4. Find the number of subsets of {1, ...
Packings with large minimum kissing numbers
... It would be interesting to decide if there is a linear binary code of length n in which the number of words of minimum Hamming weight is 2Ω(n) . It would also be interesting to decide if there is a lattice packing in Rn whose kissing number is 2Ω(n) . We strongly suspect that such codes and such lat ...
... It would be interesting to decide if there is a linear binary code of length n in which the number of words of minimum Hamming weight is 2Ω(n) . It would also be interesting to decide if there is a lattice packing in Rn whose kissing number is 2Ω(n) . We strongly suspect that such codes and such lat ...
Lecture Notes 2: Infinity
... The set S: the set of all and only ordinary sets, i.e. the set of all sets that are not elements of themselves. Question: Is S an ordinary set? If S is an ordinary set, then it contains itself as an element, since it is supposed to contain all ordinary sets. But that means that S is an extraordinar ...
... The set S: the set of all and only ordinary sets, i.e. the set of all sets that are not elements of themselves. Question: Is S an ordinary set? If S is an ordinary set, then it contains itself as an element, since it is supposed to contain all ordinary sets. But that means that S is an extraordinar ...
Algebra Autumn 2013 Frank Sottile 24 October 2013 Eighth Homework
... elements. (c) This action realizes S3 ≀ S2 as a sugroup of S6 . What are the cycle types of permutations of S3 ≀ S2 ? For each cycle type, how many elements of S3 ≀ S2 have that cycle type? ...
... elements. (c) This action realizes S3 ≀ S2 as a sugroup of S6 . What are the cycle types of permutations of S3 ≀ S2 ? For each cycle type, how many elements of S3 ≀ S2 have that cycle type? ...
Document
... A set A is countable if it is either finite or N is equivalent to A . Remark If an infinite set A is countable, then we can list its element as a sequence A a1 , a2 , a3 , . ...
... A set A is countable if it is either finite or N is equivalent to A . Remark If an infinite set A is countable, then we can list its element as a sequence A a1 , a2 , a3 , . ...
Language of Sets
... mathematical purposes we can think of a set intuitively, as Cantor did, simply as a collection of elements. ...
... mathematical purposes we can think of a set intuitively, as Cantor did, simply as a collection of elements. ...
Math330 Fall 2008 7.34 Let g be a non
... 7.34 Let g be a non-identity element of G. Then |g| must be 12, 6, 4, 3, or 2. If the order of g is 2 then we are done. If |g| = 12, 6, 4 then |g 6 | = 2, or |g 3 | = 2, or |g 2 | = 2. So the only way for there not to be an element of order two in G is if all non-identity elements have order 3. Let’ ...
... 7.34 Let g be a non-identity element of G. Then |g| must be 12, 6, 4, 3, or 2. If the order of g is 2 then we are done. If |g| = 12, 6, 4 then |g 6 | = 2, or |g 3 | = 2, or |g 2 | = 2. So the only way for there not to be an element of order two in G is if all non-identity elements have order 3. Let’ ...
DISCRETE MATH: LECTURE 20 1. Chapter 9.2 Possibility Trees
... 1. Chapter 9.2 Possibility Trees and the Multiplication Rule continued... Theorem 1.1 (Equally Likely Probability Formula). If S is a finite sample space with N (S) elements in which all outcomes are equally likely and E ⊆ S is an event with N (E) elements, then the probability of E is: ...
... 1. Chapter 9.2 Possibility Trees and the Multiplication Rule continued... Theorem 1.1 (Equally Likely Probability Formula). If S is a finite sample space with N (S) elements in which all outcomes are equally likely and E ⊆ S is an event with N (E) elements, then the probability of E is: ...
Birkhoff's representation theorem
This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.