5. True or False. Explain your answers. (a) If every element of G/H has finite order, and every element of H has finite order, then every element of G has finite order. (b) If |G/H|= =p, then the order of every element a H in G is a multiple of p.

Q: Let G= Z* and let * be an operation in a * b. Is a * b = ab/2 is closed under the operation?

A: Disproof by an example.

Q: If 0 ≤ an ≤ b, and Σb, diverges, then Σan diverges.

A: Given: The series an and bn in the interval 0≤an≤bn and if the series ∑bn diverges, then ∑an…

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A: Given that S= {1,2,3}×{0,1,2,3,4} We need to find the subset of S.

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A:

Q: 2.47. Let P(x) be the assertion "x is odd", and let Q(x) be the assertion "x2 - 1 is divisible by…

A: Disclaimer: Since you have asked multiple question, we will solve the first question for you. If you…

Q: Which relation below is a total order? O A = {2,3,4} ordered by the relation R on A defined as…

A:

Q: For any two non-negative integers x and y, let d(x, y) = x + y. Let x, y, z be any three…

A: See solution below

Q: EXAMPLE 0.7 The set N² equals Nx N. It consists of all ordered pairs of natural numbers. We also may…

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Q: 1 2 1 Let f= 34 5 6 7 8 9 2 7 8 4 and g = 2 3 4 5 6 7 8 9 4 3 8 1 52 6 be permutations in 6 9 5 1 3…

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Q: Find the maximum possible order for some element of Z, x Z1, x Z18 O 50 O 150 O 90 O 80 None of them

A: Given problem is :

Q: n S = {3, 5, 7, 8, 9, 10, 12, 27, 30, 70} we work with the divisibili xRy ⇒ x divides y. (i) Draw…

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Q: How do you derermine of RS → R* 76 onto?

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Q: 2.6.2(1) Let A CR be a non – empty set. Refer to Example 2.5.13 for the definition of a greatest…

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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: C. Draw a Venn Diagram showing the number of students in each of the eight sections of the Venn…

A: Note: as per guidelines we support only three subparts of the questions.kindly repost next question…

Q: 1.5.1 Prove that the following sets are uncountable. (a) A is the set of all sequences r = Tk is an…

A: We have to prove following

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Q: 3. Find the cube root of the given complex number: y= (4 – j3) i.e., find V(4 – j3)!. You may leave…

A: Given: z=4-3j To find: Cube root of a given complex number.

Q: 2. Let f and g be any permutations. Show that the order of h = fgf- is the same as the order of g.

A: Here f and g are tow permutations  and we are given that h=fgf-1. We have to show that both f and g…

Q: Let S = ((a), b, (a, (b)]), c). Which one of the following relations on S is a strict partial order?…

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Q: 2. List all elements of A = M₂ (Z₂). Give the units in A, as well as the zero divisors in A.

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Q: Question B3: Let A = {a,b}, B = {1,2,3}, and C = {r, s}. (a) Use the roster method to list the…

A: Given sets are  A= a,b ,  B=1,2,3 ,  and  C=r,s   (a)  A×B×C = a,1,r, a,1,s, a,2,r, a,2,s, a,3,r,…

Q: A7. Jacobsen has a box with 2000 white balls. He also have an unlimited supplies of white, green,…

A: Solution: Part(a) After finitely many operations there are three balls left in the box. At least one…

Q: 3. Let X be the set of positive integers. For each n E X, let Sn = {k € X: k2n}. Show that T = {S:…

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Q: 2. Consider S3 = the set of all permutations on the set [3]. You can find these 6 permutations in…

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

Q: Find the order of each element in U(15).

A:

Q: Let S-(0(3).(3.61(3.61).(3.6.9 12) Then the poset (S) is None of these not a total order set because…

A: Total order Set -:  A Poset (A,~) is called total order if any two elements are comparable, i,e. if…

Q: Question 11 Given domain of the relation R is {a}. R = {(a, a)} is a reflexive, symmetric and…

A: The domain of relation R is a.

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A: We can calculate the permutation of f7 from right to left then take the LCM of disjoint cycle

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Q: (6) Suppose A CR is countable. Show that there exists a real number r such that An(r+ A) = 0. (Here,…

A: We prove this by contradiction

Q: (3) Let G = ( di [pl 6789 3 6 5 1 4 9 8 7 2 (a) Compute c³ and or 1 2 3 4 5 Is Is σ (b) Worte o…

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Q: 8. All polynomials in P, such that p(0) = 0.

A: