1 Sets
... Sometimes it is difficult to prove directly a statement and is also difficult to find a counterexample. There are statements in mathematics (called open problems) that we know they are sure either true or false, but not both. However, we are just neither able to prove the statement nor to find a cou ...
... Sometimes it is difficult to prove directly a statement and is also difficult to find a counterexample. There are statements in mathematics (called open problems) that we know they are sure either true or false, but not both. However, we are just neither able to prove the statement nor to find a cou ...
Solutions - UBC Math
... both 3 − 2i and 3 + 2i are irreducible, since their norms are equal to 13, which is prime; and we know that (a) Norm is multiplicative, so if we had 3 − 2i = αβ, then N (3 − 2i) = N (α)N (β), so this forces N (α) or N (β) to be 1; and (b) Any element of norm 1 is a unit. (b) Prove that any ideal in ...
... both 3 − 2i and 3 + 2i are irreducible, since their norms are equal to 13, which is prime; and we know that (a) Norm is multiplicative, so if we had 3 − 2i = αβ, then N (3 − 2i) = N (α)N (β), so this forces N (α) or N (β) to be 1; and (b) Any element of norm 1 is a unit. (b) Prove that any ideal in ...
1 Sets, functions and counting
... A set is a well defined collection of distinct definite objects. The objects that make up the set are its elements. If S is a set and s is an element of S, we write s ∈ S. On the other hand if s is not an element of S then we write s 6∈ S. To specify a set, list its elements separated by commas and ...
... A set is a well defined collection of distinct definite objects. The objects that make up the set are its elements. If S is a set and s is an element of S, we write s ∈ S. On the other hand if s is not an element of S then we write s 6∈ S. To specify a set, list its elements separated by commas and ...
C SETS - UH - Department of Mathematics
... 1. The set of vowels in the word “probability” is the set C = {a, o, i}. Note that the letter i appears twice in the word “probability” but we list it once - a set is a collection of distinct elements. 2. The set of real numbers that satisfy the equation x 2 − 9 = 0 is the set S = {− 3,3} . 3. The s ...
... 1. The set of vowels in the word “probability” is the set C = {a, o, i}. Note that the letter i appears twice in the word “probability” but we list it once - a set is a collection of distinct elements. 2. The set of real numbers that satisfy the equation x 2 − 9 = 0 is the set S = {− 3,3} . 3. The s ...
ON A LEMMA OF LITTLEWOOD AND OFFORD
... sum ]Qfc=i€fcxfc w e associate a subset of the integers from 1 to n as follows: k belongs to the subset if and only if e&= + 1 . If two sums 2^J»i€^jb and ]Cfc=i*& #& are both in 7, neither of the corresponding subsets can contain the other, for otherwise their difference would clearly be not less t ...
... sum ]Qfc=i€fcxfc w e associate a subset of the integers from 1 to n as follows: k belongs to the subset if and only if e&= + 1 . If two sums 2^J»i€^jb and ]Cfc=i*& #& are both in 7, neither of the corresponding subsets can contain the other, for otherwise their difference would clearly be not less t ...
Exam I Solutions Topology (Math 5863) 1(a) If X and Y are
... find a neighborhood of (x, f ) that is mapped by e into V . Since f is continuous and f (x) ∈ V , the set W = f −1 (V ) is an open neighborhood of x in X. By local compactness and the Hausdorff property, there is a neighborhood U of x such that U ⊂ W and U is compact. Now U × S(U , V ) is an open ne ...
... find a neighborhood of (x, f ) that is mapped by e into V . Since f is continuous and f (x) ∈ V , the set W = f −1 (V ) is an open neighborhood of x in X. By local compactness and the Hausdorff property, there is a neighborhood U of x such that U ⊂ W and U is compact. Now U × S(U , V ) is an open ne ...
Recently Littlewood and Offord1 proved the following lemma Let x1
... sum zL= 1 Ek :Y'k we associate a subset of the integers from 1 to n as follows : k belongs to the subset if and only if ek= +1 . If two sums ~k_,Ekxk and Zx=1E1 . xk are both in I, neither of the corresponding subsets can contain the other, for otherwise their difference would clearly be not less th ...
... sum zL= 1 Ek :Y'k we associate a subset of the integers from 1 to n as follows : k belongs to the subset if and only if ek= +1 . If two sums ~k_,Ekxk and Zx=1E1 . xk are both in I, neither of the corresponding subsets can contain the other, for otherwise their difference would clearly be not less th ...
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.