What is the cardinality of the set of 2 x 2 matrices over N? Prove your answer.  Note: a 2 x 2 matrix over N is a 2 x 2 matrix whose elements are all natural numbers, for example  1   2                17    82 3   4      and     1      1  are both 2 x 2 matrices over N

Q: Compute f2,f3, and f-1 for each of the following permutations.   f= (1,5,2,4)

A: Given: f=(1,5,2,4) To determine the f2 f3 f-1

Q: 3. Prove that (An B)² C A × B.

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Q: Show that U(8) and U(10) are not isomorphic.

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Q: Let &={2,0, E,E), then A is not o – algebra.

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Q: Exercise A.2.17. (1) Show that gcd(a,n) = 1 if and only if there exist m and k such that ma + kn =…

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Q: 6. Prove that if gcd(a , b) = 1, then gcd(a + b, ab) = 1.

A: We need to prove that if gcda,b=1 then gcda+b,ab=1 Let us on the contrary assume that  gcda+b,ab=c…

Q: 2. Show that the following is a basis for M, 2x2 [3 6 3 1 0 -1 2 -8 12

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Q: If A ánd B are two n x n invertible matri hen one of the following is always correct

A: It is given that,  A and B are two n×n invertible matrices.

Q: 3. If x is uncountable, then prove that A = {ECX: ECX: algebra O X: E is countable or E² is…

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Q: Prove that if gcd(a, b) = 1, then gcd(a + b, ab) = 1.

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Q: m a basis for R° if and only if k +

A: Here we have,  v1→=7-50, v2→=-335, v3→=12-6k form a basis for R3 if and only if k≠?

Q: Prove that if a eZ then the only positive divisor of both a and a+1 is 1.

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Q: 3E.1 Let H(q) = 1 – 1.1q¬ + 0.34-2 Compute H-(q) as an explicit infinite expansion.

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Q: Let p be a primes. if the linear system X a (mod p) x = b (mod qp) is solvable then a-b =0 (mod p)…

A: The given statement is TRUE.

Q: Find bases for the null space and row space of A. 5. 8. -7 -4 -1 -8 1 3. 3 A = -1 0. -1 1 -2 4 7. 10…

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Q: Exercise 16: Write up three proofs for the above six theorems. (You should be able to do all of…

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Q: Prove that if gcd (a,b)=1>then gcd (ab,ab)=l

A: I have proved using the fact that gcd can be expressed as integers linear combination q

Q: Compute f2,f3, and f-1 for each of the following permutations. f= (2, 7,4,3,5)

A: Given: f= (2, 7,4,3,5) To Compute: f2,f3, and f-1

Q: Find the norm of the partitionP = {−5,−4.3,−3.2,−2.3,−1.8,−1}

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Q: Let S = {(1, 0, i), (1, 2, 1)} in C3. Compute S⊥.

A: For the given space in C3 let the desired perpendicular space be (a,b,c). It follows the conditions…

Q: Suppose that 2 = {1,2,3,4} and C = {{1}, {2}, {3,4}}. Determine o(C), the smallest o-algebra on 2…

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Q: A diet requires 4200 calories, 900 grams of food, and 80% of the Calories from carbohydrates and…

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Q: Prove that if m | k then 2" – 1| 2k – 1

A: Use the definition of Divisibility, "a divides b" means a and b are integers and there is an integer…

Q: Let f and g be permutations. Show that the order of h=fgf^−1 is the same as the order of g.

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Q: (e) There exist subsets A and B of N such that (A-B) n (ANB) = TRUE FALSE

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Q: consider the following arrays defined in Matlab's workspace z=[2 9 0 ; 7 5 6] Retrieve or create…

A: Given,

Q: Prove that |A U B| + |A n B| = |A| + |B|

A: Let A and B are two sets. A = number of elements in the set A. B = number of elements in the set B.…

Q: *prove that for . P+2+3+... +n° = 2 n=!,2,3,

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

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

Q: 7.21 A set of five codewords for a vector quantization algorithm is shown below. Draw the cells…

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Q: Prove or disprove that if {a,} and {b,} are Cauchy and b,n # 0. Then An is Cauchy

A: If an and bn are Cauchy and bn≠0, then anbn is not Cauchy. Result used: If an and bn are Cauchy and…

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: 6. Prove that if gcd(a, b) 1, then gcd(a +b, ab) = 1. %3D %3D

A: Given that gcd(a,b)=1 If possible , let us assume that  gcd(a+b,ab)=d,d≠1 Then the number d must…

Q: Let m and n be two relatively prime positive naturals, and consider what naturals can be expressed…

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Q: Prove Theorem . k(AB) = {kA)B = A{kB)

A: Proof: Given:   Let A and B are matrices of the same order and let k be the scalar.  To prove:…

Q: Prove that if gcd(a, b) = 1 and gcd(a, c) = 1, then gcd(a, bc) = 1. %3D

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Q: Prove for all m, n, and k in w, that (m⋅n)k = mk.nk.

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Q: 4. Assuming that gcd(a, b) = 1, prove the following: (a) gcd(a +b, a – b) = 1 or 2. %3D %3D

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: (a) Suppose that V =Span{ 2 2 8 Determine whether each of the following sets is a basis for V: 2 | 2…

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Q: Consider the Poset (Z+, |). The greatest lower bound of {2, 3, 6, 9, 12} is : a. 1 b. 0 О с. O d.…

A: Given set is 2,3,6,9,12 We have to find the greatest lower bound of the set 2,3,6,9,12 Definition:-…

Q: Find the norm of the partitionP={1,1.1,1.7,2.4,2.9,3.1,3.8,4}

A: Given Partition, To find the Norm of Partition.

Q: I need the answer as soon as possible

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Q: Let A = 1 2 1 and v = If the entries of A and v belong to Z3, what is the third entry of Av? Answer:

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Q: If 2%= N₁, then a Ramsey ultrafilter exists.

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Q: Prove that vn e Z, gcd(7n + 1, 15n + 2) = 1 *gcd = Greatest common divisor

A: Below is the solution for the above question.

Q: Theorem 26. If n ∈ N then (1^n) = 1. Theorem 27. If x, n, m ∈ N then (x^m)^n = x^(mn) . Exercise…

A: Given 1n =1 where n belongs to N(natural numbers)   Prove .. 1×1×1×1.........×1 n Times.........(1)…

Q: linear algebra, 2.5 question 5

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Q: (b) Show that if m < n, then any m x n such array can be extended to a (m + 1) x n array satisfying…

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