Consider the homomorphism : Z45 → Z45 defined by p(x) = 5x mod 45. (a) List the elements of Kero. (b) Given that (20) = 10, list all elements x € Z45 satisfying (x) = 10.

Q: Prove directly that there cannot exist a homomorphism : Q8 that (j) = (1234) and (k) = (5314). S5…

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Q: 1- Which of the following is a homomorphism? a)O f:Z→Z, f(x) = x+2 %3D b) O f: Z12 → Z30, ƒ(x) = 20x…

A: Consider a function    .Then f is a homomorphism if and only if  . .

Q: An example of a partition of the integers is E Z}

A: Set of integers is defined as {... -3, -2, -1, 0, 1, 2, 3,....}. A partition of a set is defined as…

Q: Let X8 = {1, 2, ..., 8}, and let Y8 Yε = {b₁ b₂b3b4b5b6b7b8 | b; = {0, 1}} be the set of binary…

A: In this problem, we are given two sets:Set X8={1,2,...,8}, which consists of the integers from 1 to…

Q: Select the following relations on Z that are equivalence relations. O{(a,b)|a>b-1} {(a,b) | a² = 6²…

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Q: Using the natural homomorphism(which takes every integer to its equivalence class) of Z onto Zk for…

A: The idea is to reduce the equation mod 5

Q: Let G = {[1 0 0 1] ,[−1 0 0 1] ,[1 0 0 −1], [−1 0 0 −1]} . Is G ∼= K4? If yes, give an explicit…

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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: Prove that in a Boolean Algebra, x.ỹ = 0 if and only if x. y = x .

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Q: Show that every element of GF(pn) can be written in the form ap forsome unique a in GF( pn).

A: In this question, we have to show that, every element of GF(pn) can be written in the form ap for…

Q: The mapping : (Z6, +6) → (U4, X4) on my avong bas 0 is arounico at wey song has 18 uniton ai n3"…

A: Mapping :

Q: Let n = 2k with k ≥ 2. (a) Prove that pk is an element of order 2 which commutes with all elements…

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Q: Prove that π2 -1 is algebraic over Q(π3).

A: We have to prove that π2-1 is algebraic over Qπ3.

Q: (a) Explain why any automorphism (b) Show that any automorphism that Aut (S3) 6. of S3 takes a…

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Q: Define L:Z → Z and M:Z → Z by the rules L(a) = a², M (a) = a mod 5, V a ɛ Z. Find the values of (a)…

A: Composition of two functions f(x) , g(x)  is defined as : (f∘g)(x)=f(g(x))         If we have to…

Q: Determine whether the following generalizations are true or false. 1.  If x and y are both elements…

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Q: Find the inverse of 1 + √2 in Z[√2]. What is the multiplicativeorder of 1 + √2?

A: Here ℤ2=a+b2/a, b∈ℤ We need to find multiplicative inverse of 1+2 in  ℤ2. An element b∈R is said to…

Q: give a combinatorial proof of the identity  P(n,k) * (n-kCm-k) = (nCm) * P(m,k) where all…

A: Detailed proof is provided in the next step.

Q: 6. Given the following relations on the set of all integers where (z,y) = R if and only if the…

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Q: (11) Let Z, denotes the integers (mod n), e.g., Z,, denotes the 10 equivalence classes Z1, = {[0],…

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Q: 3. Show that Z10 is isomorphic to Z2 x Z5, but that Zg is not isomorphic to Z2 × Z4.

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Q: Show that φ(n2) = nφ(n) for every n ∈ Z+.

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Q: Let A = {1, 2, 3, 4} and consider the relation R on A given by aRb if and only if a−b ≡ 0 mod 4, or…

A: Given: The set A=1, 2, 3, 4, and the relation R:A→A as  R=a,b: a-b≡0mod4 or a-b≡1mod4 or a-b≡-1mod4…

Q: 2. List all the polynomials in Z3[x] that have degree 2.

A: To find all the polynomials in Z3[x] that have degree 2

Q: Prove or disprove the relation C on Z defined by C = {(x, y): x=y (mod 5)} satisfies the two…

A: I just used the definition of congruence and divisibility