Complete all parts. Do not skip any portion of the question.
Identify which of the following are rings. List out of the properties. For those that are not rings, explain why not. Assume addition and mulitiplication unless stated otherwise.
For those that are rings, state the identity elements under both operations. Note if the ring is noncommutative. List the units (those with multiplicative inverses) of each ring.
![a) Mn(Z[]). b) Z × R, with (a, b) + (c, d) = (a + c, b + d), (a, b) , (c, d) = (ac, bd). c) R x R, with (a, b) (c, d) = (a+ c, b + d), (a, b) , (c, d) = (ac-bd, ad+be) d) R × R, with (a, b) + (c, d) (a + c, b+ d), (a, b) . (c, d) = (ac-bd, ad+be). e) aRx], the polynomials that are multiples of a. f) xR[x], the polynomials that are multiples of x. i) {continuous functions f : R → R} under addition and composition.](http://img.homeworklib.com/questions/d0efebc0-570d-11eb-bd68-adb07e7fa995.png?x-oss-process=image/resize,w_560)
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Complete all parts. Do not skip any portion of the question.
Identify which of the following...
1. Let Q be the set of polynomials with rational coefficients. You may assume that this...
1. Let Q be the set of polynomials with rational coefficients. You may assume that this is an abelian group under addition. Consider the function Ql] Q[x] given by p(px)) = p'(x), where we are taking the derivative. Show that is a group homomorphism. Determine the kernel of 2. Let G and H be groups. Show that (G x H)/G is isomorphic to H. Hint: consider defining a surjective homomorphism p : Gx HH with kernel G. Then apply the...
please answer all the questions. question 1 to question 5 Given an integral domain R we...
please answer all the questions.
question 1 to question 5
Given an integral domain R we define the relatic n~on Rx (R (0]) by (a, b)~(c, d) means ad bc. We also define the following operations on R x (R\o) (a, b) + (c, d) (ad + be, bd) and (a, b) (c,d) (ac, bd). 1. Prove that ~ is an equivalence relation. 2. Prove that ~is compatible with +and . (Therefore, ~is a congru- 3. Conclude that the following...
Are the following relations in BCNF? 3NF? (if R is not in BCNF, decompose it to...
Are the following relations in BCNF? 3NF? (if R is not in BCNF, decompose it to BCNF; if R is not in 3NF, decompose it to 3NF) R(X,Y, Z,T,V): XY->Z, Y->T, Z->V R(X,Y,Z,T): X->Y, Y->Z, Z->T R(A,B,C): AB->C, B->A, C->B R(ABCD): BD->C, AB->D, AC->B, BD->A R(ABCD): AD->C, CD->B, BD->C R(ABCD): A->C, B->A, A->D, AD->C R(ABCD): A->D, C->A, D->B, AC->B R(XYZT): XYT->Z, ZT->X, XZ->Y, XZ->T R(XYZT): XY->Z, XYT->Z, XYZ->T, XZ->T R(XYZT): YT->Z, XY->T, XZ->Y, YT->X R(XYZT): YZ->X, XT->Z, ZT->Y, YT->Z
Please answer all parts. Thank you! 20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R...
Please answer all parts. Thank you!
20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R × S by (a, s) ~ (a, s') if there exists an s"e S such that s* (s,a-sa,) a. 0. Show that ~ is an equivalence relation on b. Let a/s denote the equivalence class...
Numbers 3,4,11 a. SublactiTlnb b. division of nonzero rationals c. function composition of polynomials with real...
Numbers 3,4,11
a. SublactiTlnb b. division of nonzero rationals c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with integer entries e. exponentiation of integers 3. Which of the following binary operations are commutative? a. substraction of integers b. division of nonzero real numbers c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with real entries e. exponentiation of integers 4. Which of the following sets are closed...
7 Which of the following could you conclude using coordinate geometry? A. AEFG is an equilateral...
7 Which of the following could you conclude using coordinate geometry? A. AEFG is an equilateral triangle. B. m.E = 120 C. m F = 99 D. m.E= m F. ОА OB Ос Given: Parallelogram ABCD Prove: AC bisects BD, and BD bisects AC. Plan: Place the parallelogram in the coordinate plane with a vertex at the a._? and a side along the b._? . Since midpoints will be involved, use multiples of c. ? to name coordinates. To show...
Which of the following mappings are homomorphisms? Monomorphisms? Epimorphisms? Isomorphisms? a) ...
Which of the following mappings are homomorphisms? Monomorphisms? Epimorphisms? Isomorphisms? a) G= (R-(0), .), H= (R+,-); φ: G→H is given by φ(x)=국 b) Ga(R" ,-); ф: G-* G is given by ф(x)-Vx c) G-group of polynomials p(x) with real coefficients, under addition of polynomials; φ: G-→(R, +) is given by φ[P(x)]-PC) d) G is as in (c); ф: G-G is given by ф[P(x)-p'(x), the derivative of P(x) e) G-the group of subsets of {1,2,3,4,5) under symmetric difference; A-(1,3,4), and p:...
Let F49 be the field of 49 elements constructed in class. The definition of this field...
Let F49 be the field of 49 elements constructed in class. The definition of this field is F19={la(x)]F: a(r) e Z,a}} where Z7]is the ring of polynomials in r with coefficients in the field Z7 and a(x)p = {a(x)+ (1]zz + [4],)5(x) : 5(#) e Z7(a]} and addition is given by [a(r)]F+ [b(r)]F = [a(r) + b(2)]F and multiplication is given by [a(r)]F[b(x)]F = [a(z)b(1)]p. 1. Let Fa9t represent the ring of polynomials with coefficients in F9 (a) Show that...
5. For the following set of Fischer projections answer each of the questions below by circling...
5. For the following set of Fischer projections answer each of the questions below by circling the appropriate letter(s) or letter combination(s). Hint: Redraw the Fischer projections with the longest carbon chain in the vertical direction and having similar atoms in the top and bottom portion. Classify all chiral centers in the first structure as R or S absolute configuration. (X pts) a. Which are optically active? b. Which are meso? c. Which is not an isomer with the others?...
i cant seem to figure out 6.16 Diophantus passed à of his life in childhood, in...
i cant seem to figure out 6.16
Diophantus passed à of his life in childhood, in youth, and more as a bachelor. Five years after his marriage was born a son who died 4 before his father, at his father's [final] age."How old was years Diophantus when he died? (b) Solve the following problem, which appears in Diophantus' Arithmetica (Problem 17, Book I): Find 4 numbers, the sum of every arrangement 3 at a time being given; say, 22, 24,...
1. Let Q be the set of polynomials with rational coefficients. You may assume that this is an abelian group under addition. Consider the function Ql] Q[x] given by p(px)) = p'(x), where we are taking the derivative. Show that is a group homomorphism. Determine the kernel of 2. Let G and H be groups. Show that (G x H)/G is isomorphic to H. Hint: consider defining a surjective homomorphism p : Gx HH with kernel G. Then apply the...
please answer all the questions.
question 1 to question 5
Given an integral domain R we define the relatic n~on Rx (R (0]) by (a, b)~(c, d) means ad bc. We also define the following operations on R x (R\o) (a, b) + (c, d) (ad + be, bd) and (a, b) (c,d) (ac, bd). 1. Prove that ~ is an equivalence relation. 2. Prove that ~is compatible with +and . (Therefore, ~is a congru- 3. Conclude that the following...
Please answer all parts. Thank you!
20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R × S by (a, s) ~ (a, s') if there exists an s"e S such that s* (s,a-sa,) a. 0. Show that ~ is an equivalence relation on b. Let a/s denote the equivalence class...
Numbers 3,4,11
a. SublactiTlnb b. division of nonzero rationals c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with integer entries e. exponentiation of integers 3. Which of the following binary operations are commutative? a. substraction of integers b. division of nonzero real numbers c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with real entries e. exponentiation of integers 4. Which of the following sets are closed...
7 Which of the following could you conclude using coordinate geometry? A. AEFG is an equilateral triangle. B. m.E = 120 C. m F = 99 D. m.E= m F. ОА OB Ос Given: Parallelogram ABCD Prove: AC bisects BD, and BD bisects AC. Plan: Place the parallelogram in the coordinate plane with a vertex at the a._? and a side along the b._? . Since midpoints will be involved, use multiples of c. ? to name coordinates. To show...
Which of the following mappings are homomorphisms? Monomorphisms? Epimorphisms? Isomorphisms? a) G= (R-(0), .), H= (R+,-); φ: G→H is given by φ(x)=국 b) Ga(R" ,-); ф: G-* G is given by ф(x)-Vx c) G-group of polynomials p(x) with real coefficients, under addition of polynomials; φ: G-→(R, +) is given by φ[P(x)]-PC) d) G is as in (c); ф: G-G is given by ф[P(x)-p'(x), the derivative of P(x) e) G-the group of subsets of {1,2,3,4,5) under symmetric difference; A-(1,3,4), and p:...
Let F49 be the field of 49 elements constructed in class. The definition of this field is F19={la(x)]F: a(r) e Z,a}} where Z7]is the ring of polynomials in r with coefficients in the field Z7 and a(x)p = {a(x)+ (1]zz + [4],)5(x) : 5(#) e Z7(a]} and addition is given by [a(r)]F+ [b(r)]F = [a(r) + b(2)]F and multiplication is given by [a(r)]F[b(x)]F = [a(z)b(1)]p. 1. Let Fa9t represent the ring of polynomials with coefficients in F9 (a) Show that...
5. For the following set of Fischer projections answer each of the questions below by circling the appropriate letter(s) or letter combination(s). Hint: Redraw the Fischer projections with the longest carbon chain in the vertical direction and having similar atoms in the top and bottom portion. Classify all chiral centers in the first structure as R or S absolute configuration. (X pts) a. Which are optically active? b. Which are meso? c. Which is not an isomer with the others?...
i cant seem to figure out 6.16
Diophantus passed à of his life in childhood, in youth, and more as a bachelor. Five years after his marriage was born a son who died 4 before his father, at his father's [final] age."How old was years Diophantus when he died? (b) Solve the following problem, which appears in Diophantus' Arithmetica (Problem 17, Book I): Find 4 numbers, the sum of every arrangement 3 at a time being given; say, 22, 24,...
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