![arXiv:math/9911224v2 [math.GT] 9 Dec 1999](http://s1.studyres.com/store/data/016370695_1-31d422643fefeca7faefc96120ceb1d7-300x300.png)
arXiv:math/9911224v2 [math.GT] 9 Dec 1999
... . . . ∪ {xk }. The quotient topology gives expk X the structure of a topological space. Clearly, exp1 X = X for any X. The simplest non-trivial example is provided by the space exp2 S 1 which is easily seen to be homeomorphic to the Möbius band. The space exp3 S 1 is described by a well-known theor ...
... . . . ∪ {xk }. The quotient topology gives expk X the structure of a topological space. Clearly, exp1 X = X for any X. The simplest non-trivial example is provided by the space exp2 S 1 which is easily seen to be homeomorphic to the Möbius band. The space exp3 S 1 is described by a well-known theor ...
Proposition: The following properties hold A ∩ B ⊆ A, A ∩ B ⊆ B, A
... Definition: If A is a finite set we write |A| for the number of elements in A and call |A| the cardinality of A. Proposition: If A and B are finite sets then |A ∪ B| = |A| + |B| − |A ∩ B| Proof: From a Venn diagram we see |A ∪ B| = |A∩ ∼B| + |A ∩ B| + | ∼A ∩ B| The first two terms give |A| while the ...
... Definition: If A is a finite set we write |A| for the number of elements in A and call |A| the cardinality of A. Proposition: If A and B are finite sets then |A ∪ B| = |A| + |B| − |A ∩ B| Proof: From a Venn diagram we see |A ∪ B| = |A∩ ∼B| + |A ∩ B| + | ∼A ∩ B| The first two terms give |A| while the ...
Distributive Property
... expression. 2) Write an equivalent expression by applying the distributive property. ...
... expression. 2) Write an equivalent expression by applying the distributive property. ...
Math 3121 Lecture 6 ppt97
... Theorem: Let G be a group. The intersection of a set of subgroups of G is a subgroup of G. Proof: Let T be a set of subgroups of G, and let H be the intersection of all members of T. Then h is in H if and only if h is in K for all K in T. Now we show that H satisfies the conditions in the theorem fo ...
... Theorem: Let G be a group. The intersection of a set of subgroups of G is a subgroup of G. Proof: Let T be a set of subgroups of G, and let H be the intersection of all members of T. Then h is in H if and only if h is in K for all K in T. Now we show that H satisfies the conditions in the theorem fo ...
MTH 098
... • An algebraic expression consists of 1. variables with “counting number” exponents 2. coefficients 3. constants 4. arithmetic operations and grouping symbols • An expression will not have an equal sign. • To simplify an algebraic expression: 1. Apply the distributive property to remove parentheses. ...
... • An algebraic expression consists of 1. variables with “counting number” exponents 2. coefficients 3. constants 4. arithmetic operations and grouping symbols • An expression will not have an equal sign. • To simplify an algebraic expression: 1. Apply the distributive property to remove parentheses. ...
Counting Sets - MIT OpenCourseWare
... (denoted A-B) can be computed by subtracting A∩B from A, i.e. |A-B| = |A| - |A∩B|. Example: Let A be the set of poker hands which have all five cards the same suit and let B be the set of poker hands which contain a consecutive sequence in rank order (e.g. 3,4,5,6,7 not necessarily the same suit). T ...
... (denoted A-B) can be computed by subtracting A∩B from A, i.e. |A-B| = |A| - |A∩B|. Example: Let A be the set of poker hands which have all five cards the same suit and let B be the set of poker hands which contain a consecutive sequence in rank order (e.g. 3,4,5,6,7 not necessarily the same suit). T ...
Axioms for high-school algebra
... (F5): Multiplication is associative. Let x, y and z be any numbers. Then (xy)z = x(yz). (F6): There is a multiplicative identity. The number 1 (one) is the multiplicative identity. This means that 1x = x = x1 for any number x. (F7): Each non-zero number has a unique multiplicative inverse. For each ...
... (F5): Multiplication is associative. Let x, y and z be any numbers. Then (xy)z = x(yz). (F6): There is a multiplicative identity. The number 1 (one) is the multiplicative identity. This means that 1x = x = x1 for any number x. (F7): Each non-zero number has a unique multiplicative inverse. For each ...
Math_Practices_HS Sample_Problems
... Standards for Mathematical Practices: Sample Problems HS.N-VM.12. Work with 2 2 matrices as transformations of the 4. Model with mathematics. plane, and interpret the absolute value of the determinant in terms Use appropriate tools strategically. of area. ...
... Standards for Mathematical Practices: Sample Problems HS.N-VM.12. Work with 2 2 matrices as transformations of the 4. Model with mathematics. plane, and interpret the absolute value of the determinant in terms Use appropriate tools strategically. of area. ...
Set Theory Basic Concepts and Definitions The Importance of Set
... and not another. Many of us today like to sort our clothes at house, or group the songs on our computer into playlists. The idea of sorting out certain objects into similar groupings, or sets, is the most fundamental concept in modern mathematics. The theory of sets has, in fact, been the unifying f ...
... and not another. Many of us today like to sort our clothes at house, or group the songs on our computer into playlists. The idea of sorting out certain objects into similar groupings, or sets, is the most fundamental concept in modern mathematics. The theory of sets has, in fact, been the unifying f ...
Discrete section 1.2 Lecture
... Example 5 – Using the Set-Roster Notation a. Let A = {1, 2, 3}, B = {3, 1, 2}, and C = {1, 1, 2, 3, 3, 3}. What are the elements of A, B, and C? How are A, B, and C related? b. Is {0} = 0? c. How many elements are in the set {1, {1}}? d. For each nonnegative integer n, let Un = {n, –n}. Find U1, U2 ...
... Example 5 – Using the Set-Roster Notation a. Let A = {1, 2, 3}, B = {3, 1, 2}, and C = {1, 1, 2, 3, 3, 3}. What are the elements of A, B, and C? How are A, B, and C related? b. Is {0} = 0? c. How many elements are in the set {1, {1}}? d. For each nonnegative integer n, let Un = {n, –n}. Find U1, U2 ...
adobe pdf - people.bath.ac.uk
... (b) Suppose that G is finite of order pn where n ≥ 1, Use part (a) to show that |Z(G)| ≡ 0 mod p and deduce that |Z(G)| > 1. 2. Let G be a group of order p2 where p is a prime number. (a) Show that if G is non-abelian, we must have |Z(G)| = p and g p = 1 for every g ∈ G. (b) Suppose that G is non-ab ...
... (b) Suppose that G is finite of order pn where n ≥ 1, Use part (a) to show that |Z(G)| ≡ 0 mod p and deduce that |Z(G)| > 1. 2. Let G be a group of order p2 where p is a prime number. (a) Show that if G is non-abelian, we must have |Z(G)| = p and g p = 1 for every g ∈ G. (b) Suppose that G is non-ab ...
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.