Let 2 = {a, b, c}. Write down all possible o-algebras on N.

Q: Q1) In a ring Z, find the least common multiple of {2,3,8,12}.

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Q: (B): Let a:M → N, B: N →M are R-homomorphism such that a oB = In then M = Ker(a) + Im(B) and Ker(a)…

A: The problem is related with R homomorphisms. Given: α :M → N , β :N →M are R homomorphism, such that…

Q: be a field and let c,d e F. Show that c · (-d) = -(c d).

A: Associative Property of Field F for a,b,c∈F a·b·c=a·b·c

Q: Let ∗ be the binary operation on Z defined by x∗y=x+xy. Then (Z,∗) has a right identity element but…

A:

Q: The number of idempotents elements in the ring Zs is: O 2 4 8 1 O O O

A: Ans : 4  Option 2nd true

Q: Let ficxT)→(メッテ)and g: メっテ)→(Kっズ) e are homomorphism then gof 1s homo.

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Q: Find all possible automorphisms of Q(/2).

A: We will use basic knowledge of Galois theory and group theory to solve this question.   Note that an…

Q: Determine all ring homomorphism from Zn to Zn

A: (if n is not prime) No. of ring homomorphism from Zn to Zn = No. of Idempotent elements in Zn.

Q: Does the following equality hold in every Boolean algebra? Show all working (z'(x' + y'))' = (z+…

A: given that (z'(x'+y'))'=(z+x)(z'+z)(z+y) We need to check whether the following equality holds in…

Q: Use the class algebra to prove that if A C B and C = B – A, then A = B – C.

A:

Q: Let {1,2,3,4,5,6}. Construct the minimal field containing set A. 1. A={{1,2}, {3,4}}

A: Given: Let, Ω={1,2,3,4,5,6} A={{1,2},{3,4}}

Q: Find all ring automorphisms of Q(∛5).

A:

Q: Q1) In a ring Z, find the least common multiple of {2,3,8,12).

A: Given problem Given that   In a ring Z , We have to find the Least common multiple of { 2,3,8,12…

Q: Q) Make up a Kelly table for set G= [1,-1, i,-i} with respect to multiptication and then explain…

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Q: The number of idempotents elements in the ring Zs is: O 1 O 2 O 8 O 4

A:

Q: Let X = {1,2, 3, 4, 5, 6}, A = {0, X,{1,2,3}, {3,4}, ...} is a o-algebra on X. Which of the…

A: Topic- set and sub set

Q: A= {0, a, b, c} is a ring with four different elements having a multiplication table given in (a)…

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Q: (a) Find all the irreducible representations of S3.

A: To Determine :-  All the irreducible representations of S3.

Q: The set of all zero divisors of the ring Z6 is اختر احدى الاجابات O {2, 3, 4} O (1, 3, 5) O {1, 2,…

A: Z6={0,1,2,3,4,5}Since 2 · 3 ≡6≡ 0 (mod 6) and 3 · 4 ≡12≡ 0(mod 6)However, 1 and 5 are not zero…

Q: Find the charactes ist c of the ring Z2 Zg

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Q: Let a ER which expressions are correct? and give the justification why they are part of the Borel…

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Q: List the elements of the power set of N and enumerate all o-fields containing {0}. 1. 0 = {0} %3D 2.…

A: Sigma field is closed under countable union and complement and the empty set always belong to a…

Q: Determine A-" if A is invertible. 1 0 -2 4 -2 1 1 2 -10

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Q: Let it be a,b two elementsin thegroup Such that tafa4 and bl=2 b=ba Sind labl

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Q: Q2: Let X be anon-empty set, prove that (p(X), An) is ring?

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Q: Let Kla ab a,bE@ Show that k is closed winder addition and mulltphcatin. Show (k,ti) is a field.

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Q: Show that in a ring R cancellation laws hold if and only if R has no zero divisors.

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Q: Let X = {1,2, 3}. Which of the following is a o-algebra. 1. E, = {0, {2}, {2, 3}. {1, 2,3}}. 2. E =…

A: See solution below

Q: (1) Let A be a o-algebra in a set 2. (Give a complete definition). Take BE A. Prove that B= {ANB : A…

A:

Q: (1) (a) Define A1 := {A CR : A is open} and A2 := {A CR: A is closed}. Is Aj is a o-algebra? Is Az…

A: As per Bartleby guidelines, for more than one questions asked, only first is to be answered. Please…

Q: Find all ring homomorphisms from R to Z5.

A:

Q: Let X = {1,2, 3, 4, 5, 6}, A = {0, X, {1,2, 3}, {3,4}, ...} is a o-algebra on X. Which of the…

A: Given set is X={1,2,3,4,5,6}

Q: Let X = {1,2,3}. Which of the following is a o-algebra. 1. E, = {0, (2}, {2,3}, (1,2,3}}. 2. E = (0,…

A: Solution by using the definition of sigma algebra as follows

Q: Using only the definition of an s-algebra, show that if A and B are elements of an s-algebra, then…

A: Given that  A and B are elements of an s-algebra.

Q: The number of idempotents elements in the ring Z6 is: O 2 O 8

A: The idempotent element in the ring Z6 is 4 .   An element e in a ring R is said to be an idempotent…

Q: Let a be the set all of Let non- Zelo leal nd ab 3 dhow numbers a* b = ahouw that CG, x) abelian…

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Q: Let (L, *, O, ', 0,1) be a Boolean algebra. Then prove the following (a*b)' = a'Ð b' and (a O b)' =…

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Q: he ring 3z is isomorphic to the ring 5Z O False O True K

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Q: The set of all zero divisors of the ring Z6 is اختر احدى الدجابات O (2,3,4) O (1,3,5) O (1,2,3,4,5)…

A: The set of all zero divisors of the ring Z6 is (0,2,4)

Q: Let B be a Boolean algebra. Show that (а) ал0 — 0 (b) а V (аль) — а (c) a V (¬a A b) = a V b.

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Q: Show that there is no associative algebra A such that the Lie algebra arising from A is sl2(C).

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Q: Let a ER which expressions are correct? and give the justification why they are part of the Borel…

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Q: a) Let * be the operation on Z defined by a*b= a + b - 3ab for all a,b e Z. Justify the following…

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Q: (B): Let M be simple R -module , show if M is cyclic or not? (why)

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Q: 1 (a) Show using set identities that A UBU (AN Bn C) = AUBUC (b) Prove by contradiction that if A C…

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Q: 5. Give an example where a and b are not zero divisors in a ring R, but the sum a +b is a zero…

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Q: Let Rt st a a, b€ Z and let D=R a-4. 9. show that o E homomosphic Romemerphic Momor

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Q: Va, beZatb = a + b +2 and aob = a tabtb is a Ring!

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Q: Let a, b, c, d e R with a < b and c < d. Show that [a, b) is homeo- morphic to (c, d].

A:

Ratios

A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.

Trigonometric Ratios

Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.