1) In a group G, let a and b be two elements such that a b-ba, o(a)-4 and o(b)-5. Prove that n = {e}. Then deduce that o(ab)-20.

Q: c) Prove that the following code returns z" by structural induction func power (x,y): if (y 0):…

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Q: The permutations p= p and q in Se have the two-line symbols 1 2 3 4 5 6 4 621 5 3 and q= 1 2345 1 5…

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Q: ) (30 pts) Use induction to show the – 2" +1 _2"

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Q: In Z12, the elements [3] and [9] are * Associates Units Irreducibles None of the choices

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Q: Exercise 3. Give a combinatoric argument to prove the identity 2 2n (3.1) j=0 Hint: A subset of n…

A: Let the total persons selected be n out of 2n persons  Of which let male selected be j and the women…

Q: Let a, b, c E N. (a) Prove that if gcd(a, c) = 1 and gcd(b, c) = 1, then gcd(ab, c) = 1. (b) Prove…

A: a, b, c∈ℕ

Q: Prove that 14n is not O(12n)

A: Note that a function f(n) is big O(g(n)) if f(n) is less than constant times g(n) for large n. This…

Q: 16. Find the result of the operations indicated in each of the following problems using formulas…

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Q: Show that the k-cycle (1 2 · · · k) cannot be written as a product of fewer than k − 1…

A: We have to prove that the k-cycle (12…k) cannot be written as a product of fewer than k−1…

Q: 1. Prove that if ā and be Zn are invertible modulo n, so is ā - b. Is this the case for ā + b?…

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Q: Please use the series of numbers to answer these questions

A: Here are the random numbers given: 42718 57841 77960 73954 26526 74499We start reading the digits…

Q: (f) Write down, in cycle form, the elements of the cyclic subgroup H of So generated by p. (g) Write…

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Q: compute η2

A: η2  is given by : η2=SSbetweenSStotal

Q: Let fn be the n-th Fibonacci number of Example 3c in 8.1, k=n k=n = An Σ=1+2+. .+ n₂ =k²= 1² +2²+.…

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Q: Let a = (17 3)(5 9 4 2) and B = (2 3)(7 4)(5 18) be elements of S10. Then, a’B is O (1 4)( 2 7 5 3 9…

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Q: 2) Let A = {1,2,...,n} and let Sn denote the permutations on A. A 2-cycle of the form (a b) 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: For each divisor k>1 of n, let U,(n) = {x E U(n) | x mod k = 1}. [For example, U,(21) = {1,4, 10,…

A: For each divisor k > 1 of n , Uk(n) =x∈U(n)| xmod k =1. To Find: List the elements of U420,U520,…

Q: Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find(i) 5 ∗ 7, 20 ∗ 16.…

A: The given binary operation on N is defined as follows.

Q: Let a = (17 3)(5 49 2) and B = (2 3)(7 4)(5 1 8) be elements of Sao. Then, a²5" is O (1 279 4)(3 5…

A: No answer is given.

Q: Activity 25.22. One advantage of decomposing a permutation as a product of disjoint cycles is that…

A: The idea of permutation will be used in solving this problem

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

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Q: Remark: For positive integers n, m, we have • (n) n (m) (n) + (m) = = (lcm (n, m)) (gcd (n, m))

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Q: We have to show that nonzero elements in Zp where p is prime have multiplicative inverses

A: We have to solve given problem:

Q: Let Zn denote the set of integers {0, 1, ..., n-1}. Let be a binary operator on Z, such that a. b =…

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Q: Give a combinatorial proof that: 2n 2+()²-(²-1) = n k n k n n k=1 [Hint: There are two groups of n…

A: Given is ∑ k=1nknk2=n2n-1n-1.    We have to prove it combinatorial.

Q: 1 2 3 4 5 6 7 Let o = be a permutation in S7. Find o³4. 3 7 6 2 4 1 5 In S7, are permutations a =…

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Q: Prove or disprove: For n > 2, n − 1 is a generator of U(n)

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Q: Prove gcd(n, n+1) = 1.

A: To prove the below expression. gcdn, n+1=1