24. Let G = < C. Define 0: - by o(n) =i" Verify that o is a homomorphism a) b) Find Ker( o )

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A: This is a problem of abstract algebra.

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A: This is a problem of relation.

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A: Our objective is given below:

Q: The relation R defined on a set A is antisymmetric if (a, b) ER = (b, a) E R for (A) Every (a, b) ER…

A: A relation R is said to be antisymmetric if (a,b)∈R and (b,a)∈R implies a=b

Q: A nB oiaivalar 111222719;10s b) A'U (B n C) 9. Find the number of the subsets and list all subsets…

A: The number of subsets of a set with n elements is 2n.

Q: Find example where of is continuous and an - bijective but f¹ is not

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Q: Let x be the set of all positive integers and d(min) = | = - = | show that (x,d) is not complete

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Q: Consider the relation being described by the following arrow diagram. Which of the these statements…

A: (i) It is given that ;      S = {6, 7, 8}  and  T = {2, 4, 6} And the relation between S and T is :…

Q: is ----Llarder for - 46 8. The Ker() for the homomorphism :Z-Z₁, such that (1)=9is r 6, and r **…

A:

Q: let N be the set of natural numbers and f:NU {0} →NU [0] be definedby: (n+1 when n is even f(n) =…

A: Prove that the function f is injective for x∈N. \ The function is injective as the function is…

Q: Let A = {2" : m e Z}. A relation - is defined on the set Q+ of positive rational nu a a ~ b E A is…

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Q: Show that there exist only two homomorphism from R into R?

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Q: Let f R2 R² be an isomorphism where : → ƒ ( [¹³]) = [()] and ƒ ([×4]) = }}] Find(G)

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Q: QII State whether the following statements are true or labe: 1- Le S be anon-empty subset of the set…

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve the first three…

Q: 2. Consider morphisms f: A → B, g: BC, h: CD, k: DB, j: BC, t: A → C in a category C so that g is an…

A: Given: Morphisms f:A→B, g:B→C, h:C→D, k:D→B, j:B→C and l:A→C in a category C, where g is an…

Q: Prove relation B is an equivalence relation

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Q: The relation by a = b (modn) is Partial Order Relation Ture O False O

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Q: Describe a sequence of transformations can be used to map AJKL onto APQR? K

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Q: how to prove a map is unique?

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Q: Let R = %3 { la, b e z}and let p:R - Zbe defined by : 0(1 ) = - . 1) is a ring a) Homomorphism. b)…

A: The solution is given by using definitions of homomorphism, isomorphism and kernel as follows

Q: 5) Let R be a binary relation over the positive integers defined as follows: R = {(a, b) | 2b < a <…

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Q: For every element r in A, there is an element y in B such that (x, y) is in F. Can you please…

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Q: Def (Function): A binary relation RCX X Y is called a function from X to Y, written f:X →Y if -…

A: The concepts have been explained with examples.

Q: (D55D4+7D³ + D² - 8D + 4)y = 0

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Q: Suppose that we are interested to count the number of integers not divisible by 7, 8, 9 between 1 to…

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Q: Which of the following is isomorphism? f:(Z, +) → (Z, +) where f(x) = 2x. this option f: (R, +) →…

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Q: How many distinct equivalence classes exist in the relation R defined as below: x Ry + 3| (2x - )

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Q: R = {(x, e¹x) : x = R, n = Z}.

A: A relation is called an equivalence relation if it is reflexive, symmetric and transitive

Q: prove R is not homomorphic to R^2

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Q: 4. In the ficld (C,+, ) of complex numbers, define the mapping f: C- C by the rule f(a, b) - (a,…

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Q: 1,Y₁)~ (x2, Y₂) if x1y2 = x2Y1₁. Show t uivalence relation.

A: Given relation on the set of all ordered pairs of positive integers such that  if We have to show…