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Groups
Groups

8. Check that I ∩ J contains 0, is closed under addition and is closed
8. Check that I ∩ J contains 0, is closed under addition and is closed

... be surjective, and we get the required isomorphism. [Note that surjectivity means that for every pair (b mod m, c mod n) there exists an integer a (unique up to multiples of mn, in fact), which reduces to b mod m and to c mod n. This is exactly Chinese Remainder Theorem.] ...
A group is a non-empty set G equipped with a binary operation * that
A group is a non-empty set G equipped with a binary operation * that

Page 1 of 4 Math 3336 Section 2.1 Sets • Definition of sets
Page 1 of 4 Math 3336 Section 2.1 Sets • Definition of sets

Solutions - Cal Poly
Solutions - Cal Poly

... |a| = n and k|n, then a k = k). Similarly, if G contains an element g of order 6, then |g 3 | = 2, and if G contains an element g of order 4, then |g 2 | = 2. So we conclude that G contains an element of order 2 unless all element of G (except the identity) have order 3. This gives us exactly 11 ...
Sets and Functions
Sets and Functions

... E.g. the set of all grammatical sentences in some language Typically specified by a recursive definition Example: all positive integers not divisible by 3 ...
Combining Signed Numbers
Combining Signed Numbers

Some solutions to Homework 2
Some solutions to Homework 2

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work-pavan1
work-pavan1

... Logical implication and equivalence: If the value of Q q is true in every case in, which p is true then p is said to logically imply q, which is represented as p=>q. If p and q have same truth-value in each case then both are said to be logically Equivalent represented as p<=>q. Logical quantifies a ...
Document
Document

... We say that B is a subset of A. The notation we use is B  A. • Let S={1,2,3}, list all the subsets of S. • The subsets of S are  , {1}, {2}, {3}, {1,2}, {1,3}, ...
Distributive Property
Distributive Property

... numbers inside the parentheses by the same factor and then add the products, for instance: ...
Linear Equation System
Linear Equation System

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5.2 - shilepsky.net
5.2 - shilepsky.net

SET THEORY A set is a collection (family) of distinct and well
SET THEORY A set is a collection (family) of distinct and well

Homework 10 April 13, 2006 Math 522 Direction: This homework is
Homework 10 April 13, 2006 Math 522 Direction: This homework is

... 5. How does the subfield lattice of GF 230 compare with the subfield lattice of GF 330 ? ...
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1_7 Distributive Property

MATH 2113 - Assignment 4 Solutions
MATH 2113 - Assignment 4 Solutions

... Since this implies that x1 = x2 we have proven f is one-to-one. f (x) is not onto. If we consider the element ’1’ of the co-domain we find that if f (x) = 1 then x + 1 = x − 1 which leads to 2=0 giving a contradiction. Therefore, there is no element of the domain for which f (x) = 1. We conclude tha ...
chapter 3 part 1:sets
chapter 3 part 1:sets

A set of
A set of

... Example: Consider the class C of all open intervals of the form (a, b) on the real line. Here S  R and C  {(a, b) | a  , b  }. Consider the sequence of subsets in C given by ...
Sets and Operations
Sets and Operations

Sets and Functions - faculty.cs.tamu.edu
Sets and Functions - faculty.cs.tamu.edu

COMPARING SETS Definition: EQUALITY OF 2 SETS Two sets A
COMPARING SETS Definition: EQUALITY OF 2 SETS Two sets A

... It might help to make an analogy to something with which you are more familiar: Difference between less than (“<”) and less than or equal to (“<”) Just as “<” allows the possibility of equality: (5 < 5 is a true statement) “ ” also allows for the possibility of equality: ({1, 4, 7} ...
Assignment 4 – Solutions
Assignment 4 – Solutions

... in Z3 [x] is divided by f (x), according to the Division Algorithm. Now since deg f (x) = 3, the degree of such a remainder can be at most 2, and since its coefficients belongs to Z2 , there will be 23 = 8 such remainders, namely ...
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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.
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