1. Direct products and finitely generated abelian groups We would
... where e is the identity of H. There is an obvious correspondence between G and G0 , just send g to (e, g), and under this correspondence G and G0 are isomorphic, since e just goes along for the ride. Finally note that H × G and G × H are isomorphic; the natural map which switches the factors is an i ...
... where e is the identity of H. There is an obvious correspondence between G and G0 , just send g to (e, g), and under this correspondence G and G0 are isomorphic, since e just goes along for the ride. Finally note that H × G and G × H are isomorphic; the natural map which switches the factors is an i ...
9-2 factoring using the distributive property
... remaining factors. Then use the Distributive Property to factor out the GCF. Rewrite each term using the GCF. Simplify remaining factors. Distributive Property ...
... remaining factors. Then use the Distributive Property to factor out the GCF. Rewrite each term using the GCF. Simplify remaining factors. Distributive Property ...
On sets, functions and relations
... queen or king of the Netherlands. What is likely to be the set-notation for this set in the year 2015? 5. Describe in words the set {x ∈ Q | 0 < x < 1}. 6. Describe in words the set {x ∈ R | ∃y ∈ Q(x = y 2 )}. 7. Describe in words the set {x ∈ R | ∃y ∈ R(x = y 2 ∧ y > 2)}. 8. How many elements has t ...
... queen or king of the Netherlands. What is likely to be the set-notation for this set in the year 2015? 5. Describe in words the set {x ∈ Q | 0 < x < 1}. 6. Describe in words the set {x ∈ R | ∃y ∈ Q(x = y 2 )}. 7. Describe in words the set {x ∈ R | ∃y ∈ R(x = y 2 ∧ y > 2)}. 8. How many elements has t ...
Set Theory - The Analysis of Data
... Proof. Let An , n 2 N be a collection of countably infinite sets. We can arrange the elements of each An as a sequence that forms the n-row of a table with infinite rows and columns. We refer to the element at the i-row and j-column in that table as Aij . Traversing the table in the following order: ...
... Proof. Let An , n 2 N be a collection of countably infinite sets. We can arrange the elements of each An as a sequence that forms the n-row of a table with infinite rows and columns. We refer to the element at the i-row and j-column in that table as Aij . Traversing the table in the following order: ...
Abstract Algebra
... isomorphic, we need to show there is no one-to-one function from S onto S’ with the property (x y)= (x) ’ (y) for all x, y S. If there is no one-to-one function from S onto S’, then two are not isomorphic. This is the case precisely when S and S’ do not have the same cardinality. Recal ...
... isomorphic, we need to show there is no one-to-one function from S onto S’ with the property (x y)= (x) ’ (y) for all x, y S. If there is no one-to-one function from S onto S’, then two are not isomorphic. This is the case precisely when S and S’ do not have the same cardinality. Recal ...
I. Existence of Real Numbers
... 1 denotes the constant sequence of ones. (Conclude: there is a welldefined operation · on R that, together with the operation + obtained in 2a., makes R into a field.) 3a. Show that for any x, y ∈ C that represent distinct elements of R, the sequence x − y is either eventually positive or eventually ...
... 1 denotes the constant sequence of ones. (Conclude: there is a welldefined operation · on R that, together with the operation + obtained in 2a., makes R into a field.) 3a. Show that for any x, y ∈ C that represent distinct elements of R, the sequence x − y is either eventually positive or eventually ...
Homework 9 - Material from Chapters 9-10
... prime. (Hint: use Theorem 9.3.) Solution: Suppose for contradiction that |G : Z(G)| = p, some prime number. (So p > 1.) Now we know Z(G) C G and therefore we can consider the factor group G/Z(G). The order of this factor group equals the index, so the order is p. We know (from chapter 4 stuff) that ...
... prime. (Hint: use Theorem 9.3.) Solution: Suppose for contradiction that |G : Z(G)| = p, some prime number. (So p > 1.) Now we know Z(G) C G and therefore we can consider the factor group G/Z(G). The order of this factor group equals the index, so the order is p. We know (from chapter 4 stuff) that ...
Hoeffding, Wassily; (1953)The extreme of the expected value of a function of independent random variables." (Air Research and Dev. Command)
... the problem of the least upper bound for the expected value of the largest of n independent, identically distributed rffildom variables with given mean and variance, ...
... the problem of the least upper bound for the expected value of the largest of n independent, identically distributed rffildom variables with given mean and variance, ...
A matroid analogue of a theorem of Brooks for graphs
... Lemma 2.2. Let M be a connected binary matroid with at least two elements. Then every cocircuit of M has the same cardinality if and only if, for some positive integer t, the matroid M can be obtained by adding t − 1 elements in parallel to each element of one of the following: (i) Ur,r+1 for some r ...
... Lemma 2.2. Let M be a connected binary matroid with at least two elements. Then every cocircuit of M has the same cardinality if and only if, for some positive integer t, the matroid M can be obtained by adding t − 1 elements in parallel to each element of one of the following: (i) Ur,r+1 for some r ...
Fibonacci sequences and the spaceof compact sets
... Mathematical applications of Fibonacci-type numbers abound. In the RSA cryptosystem, for example, if an RSA modulus is a Fibonacci number, then the cryptosystem is vulnerable [Dénes and Dénes 2001]. As another example, there are no terms in the Fibonacci or Lucas sequences whose values are equal t ...
... Mathematical applications of Fibonacci-type numbers abound. In the RSA cryptosystem, for example, if an RSA modulus is a Fibonacci number, then the cryptosystem is vulnerable [Dénes and Dénes 2001]. As another example, there are no terms in the Fibonacci or Lucas sequences whose values are equal t ...
ZANCO Journal of Pure and Applied Sciences
... and used it in the way to study some properties of continuity, Based on sc open set, pc-open, bc-open and -open sets were defined in [Ameen, Z.A., 2011, Ibrahim, H.Z., 2013, Mizyed, A Y., 2015] As a generalization of sc-open,pc-open,bcopen and -open sets. In this work, we ...
... and used it in the way to study some properties of continuity, Based on sc open set, pc-open, bc-open and -open sets were defined in [Ameen, Z.A., 2011, Ibrahim, H.Z., 2013, Mizyed, A Y., 2015] As a generalization of sc-open,pc-open,bcopen and -open sets. In this work, we ...
SectionGroups
... For some sets defined over multiplication, multiplicative inverses in most cases do not exist. Example 11: Name the elements in the set of integers Z that have multiplicative inverses under multiplication. Solution: ...
... For some sets defined over multiplication, multiplicative inverses in most cases do not exist. Example 11: Name the elements in the set of integers Z that have multiplicative inverses under multiplication. Solution: ...
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.