Assignment 2, 12 Oct 2015, due 20 Oct 2015
... 1. Let L = (R, F, C) be a finite first-order relational language with F = C = ∅ and let M = (S, ι) be a finite L-structure. Show that there is an L-sentence φM whose models are precisely the L-structures isomorphic to M. 2. (a) Let L = {0, +, ×} where + and × are binary function symbols and 0 is a c ...
... 1. Let L = (R, F, C) be a finite first-order relational language with F = C = ∅ and let M = (S, ι) be a finite L-structure. Show that there is an L-sentence φM whose models are precisely the L-structures isomorphic to M. 2. (a) Let L = {0, +, ×} where + and × are binary function symbols and 0 is a c ...
1.2. Vector Space of n-Tuples of Real Numbers
... Note that some operators on either sides of these equations do not have the same meanings. For example, the + on the left of 2 denotes additions of real numbers while the + on the right denotes vector addition. Proof of the theorem is left as an exercise. ...
... Note that some operators on either sides of these equations do not have the same meanings. For example, the + on the left of 2 denotes additions of real numbers while the + on the right denotes vector addition. Proof of the theorem is left as an exercise. ...
Vector Spaces - Math Berkeley
... Given a set X and vector space V over field Fb , we may define the free space V hXi ⊂ HomF (X, V ) such that all f ∈ V hXi satisfy f (x) = 0 for all but finitely many x ∈ X. Addition and scalar multiplication are defined as usual. A crucial example of the free space is a vector field. On a manifold, ...
... Given a set X and vector space V over field Fb , we may define the free space V hXi ⊂ HomF (X, V ) such that all f ∈ V hXi satisfy f (x) = 0 for all but finitely many x ∈ X. Addition and scalar multiplication are defined as usual. A crucial example of the free space is a vector field. On a manifold, ...
notes 25 Algebra Variables and Expressions
... • Algebra is a language of symbols including variables. • A variable is a symbol, usually a letter, used to represent a number. • Algebraic expressions contain at least one variable and at least one operation. • Evaluate is to find the value of an algebraic expression by replacing variables with num ...
... • Algebra is a language of symbols including variables. • A variable is a symbol, usually a letter, used to represent a number. • Algebraic expressions contain at least one variable and at least one operation. • Evaluate is to find the value of an algebraic expression by replacing variables with num ...
BEZOUT IDENTITIES WITH INEQUALITY CONSTRAINTS
... 3. Which examples are subsets of other examples 4. Determine a basis for example 1 ...
... 3. Which examples are subsets of other examples 4. Determine a basis for example 1 ...
1. What is the cardinality of the following sets
... 3. Does A ∩ C = B ∩ C imply A = B prove your answer. 4. Show (A − B) − C ⊂ A − C. 5. Use symbolic notation to write the definition of A ⊂ B. 6. Is the function f : Z → N defined by f (x) = x2 − x one to one? Justify your answer. 7. Is the function f : students in CS247 → eyecolor defined choosing th ...
... 3. Does A ∩ C = B ∩ C imply A = B prove your answer. 4. Show (A − B) − C ⊂ A − C. 5. Use symbolic notation to write the definition of A ⊂ B. 6. Is the function f : Z → N defined by f (x) = x2 − x one to one? Justify your answer. 7. Is the function f : students in CS247 → eyecolor defined choosing th ...
A monologue - take 2? The study of Group and Ring theory is
... The identity element in abelian group (Z, +) is denoted 0. For a ∈ G, the additive inverse of a is −a. As such, −0 is the additive inverse of 0 in Z. Confusion often arises due to abuse of notation. The additive inverse in a ring can be calculated by multiplying by −1. In particular, −a = −1 ∗ a. It ...
... The identity element in abelian group (Z, +) is denoted 0. For a ∈ G, the additive inverse of a is −a. As such, −0 is the additive inverse of 0 in Z. Confusion often arises due to abuse of notation. The additive inverse in a ring can be calculated by multiplying by −1. In particular, −a = −1 ∗ a. It ...
43. Here is the picture: • • • • • • • • • • • • •
... 44. Because the polynomial is primitive, it is irreducible over Z if and only if it is irreducible over Q. As it has degree 3, it is irreducible over Q if and only if it has no roots in Q. Moreover, if a = n/d ∈ Q is a root, then n|4 and d|1, so a ∈ {±1, ±2, ±4}. Checking these one by one, we find t ...
... 44. Because the polynomial is primitive, it is irreducible over Z if and only if it is irreducible over Q. As it has degree 3, it is irreducible over Q if and only if it has no roots in Q. Moreover, if a = n/d ∈ Q is a root, then n|4 and d|1, so a ∈ {±1, ±2, ±4}. Checking these one by one, we find t ...
FINAL EXAM
... Take home final, due December 21, 2015 1. (a) Show that if p 6= q, then Qp and Qq are not isomorphic as fields. (b) Show that any field automorphism of Qp is continuous. Then show that the only automorphism of Qp is the identity. 2. We will construct number fields whose integer rings are not generat ...
... Take home final, due December 21, 2015 1. (a) Show that if p 6= q, then Qp and Qq are not isomorphic as fields. (b) Show that any field automorphism of Qp is continuous. Then show that the only automorphism of Qp is the identity. 2. We will construct number fields whose integer rings are not generat ...
1 PROBLEM SET 9 DUE: May 5 Problem 1(algebraic integers) Let K
... where p is a prime. In the former case, K is said to be of characteristic 0, while in the latter case, char K = p. (2). Let L/K be a finite extension of fields. Then L can be viewed as a finite dimensional vector space over K. Using this fact show that every finite field has order pn where p is a pr ...
... where p is a prime. In the former case, K is said to be of characteristic 0, while in the latter case, char K = p. (2). Let L/K be a finite extension of fields. Then L can be viewed as a finite dimensional vector space over K. Using this fact show that every finite field has order pn where p is a pr ...
Thinking Mathematically - homepages.ohiodominican.edu
... 4. Finally, do all additions and subtractions in the order in which they ocuur, working from left to right. ...
... 4. Finally, do all additions and subtractions in the order in which they ocuur, working from left to right. ...
Math 153: Course Summary
... Theorem 0.3 Suppose F is a finite field with N elements. Then N = pn where p is some prime. Moreover, for any number of the form pn , there is a unique field F having exactly pn elements. These finite fields have an amazing structure, and one of the goals of Math 153 is to explore it. Overall, Math ...
... Theorem 0.3 Suppose F is a finite field with N elements. Then N = pn where p is some prime. Moreover, for any number of the form pn , there is a unique field F having exactly pn elements. These finite fields have an amazing structure, and one of the goals of Math 153 is to explore it. Overall, Math ...
Introduction to Group Theory (cont.) 1. Generic Constructions of
... this case, M(S,G) with · is a group with exactly one element and the product of the map with itself is itself (by abuse of notation). Don’t consider this case when proving the following proposition. Proposition: Let (G,·) be a group and S an arbitrary set. Then (M(S,G), · ) is a group. Definition: L ...
... this case, M(S,G) with · is a group with exactly one element and the product of the map with itself is itself (by abuse of notation). Don’t consider this case when proving the following proposition. Proposition: Let (G,·) be a group and S an arbitrary set. Then (M(S,G), · ) is a group. Definition: L ...