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.] ...
... 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.] ...
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 ...
... |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
... 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 ...
... 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 ...
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 ...
... 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
... 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}, ...
... 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
... numbers inside the parentheses by the same factor and then add the products, for instance: ...
... numbers inside the parentheses by the same factor and then add the products, for instance: ...
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 ? ...
... 5. How does the subfield lattice of GF 230 compare with the subfield lattice of GF 330 ? ...
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 ...
... 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 ...
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 ...
... 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 ...
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} ...
... 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
... 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 ...
... 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 ...
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.