2 Let o (1 3 5)(2 4) and T (1 5)(2 3) be elements of S. Find o,or,to, and to

Q: 3.33 Let 2 -1 0 4 3 -1 1 6. 1 -3 -4 1 | A = and B -3 1 2 1 -4 3 0 0 1 1 -1 -1 1 Find Err(A, B) and…

A: To find Err(A, B) and PSNR(A, B) where A=2−10410−32−31210110 and B=3−11600−41−43000−1−11

Q: 2. Let A Ø. If A has exactly n elements, then A has exactly 2" subsets, Vn € N.

A: A is non empty subset of n elements ,then A has exactly 2n subsets,∀n∈ℕ.

Q: 4. In the figure, if AB = AC and BO = OC. Prove that ZB = C .C B

A: a). Given: AB=AC and BO=OC Prove: ∠B=∠C

Q: 6. let A 6 6 -3 -2 -1 -2 -5 -3 diagomol moatie over te Lield R? Oer C I Is A similar do a

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Q: (c) 2, 3, 4+ V10 and 4. V10 are irreducible elements of R.

A:

Q: 6. Let a1, a2,...,an be distinct positive in not containing s = a₁ + a₂ + + an. A grassh A AL icht…

A:

Q: 5. Prove that S(n +1, k+1) = £ $(m, k) (").

A: To prove- Sn+1,k+1=∑m=knS(m,k)nm The falling factorial is defined as the polynomial,…

Q: 10. Show that for all n E Z, n² – 3n +1 2.

A: Use mathematical induction to prove the given statement. Check whether the result is true for n=1.…

Q: Show that if all A's are B's, and no B's are C's, then no A's are C's.

A:

Q: 1. Give an explicit formula for each sn a) a, = 5an-1 – 6an-2 + 6(3")

A: The objective is to give an explicit formula for each sn an=5an-1-6an-2+63n

Q: Show that if m e Z+ and a e Z such that gcd(a, m) = 1 and ordma = m – 1, then m is prime.

A:

Q: Prove that E, k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

A: Note that there is a typo in the question. We need to show that,                            We will…

Q: Let M = {14*:k E Z*} and A = {7k:k E Z*}. Then ACM A=M O MC A O None of these

A:

Q: 2. Suppose a, b, and care integers. Prove that if a |band alethen a |b+c and a|b-c.

A: Assume a|b and a|c.

Q: d. a specific element of R° that is not in V. е. a specific element of v in V for which -v is not in…

A: Given (a,b,c) where c=a-2b

Q: If X(z) = In(z - 2), then what's x(k) а. O b. ak O C. -ak / k d. others е. ak / k

A: Z transform and inverse Z transform

Q: 1. Let S = {z € C| 26 = 1}. (a) For each element w in S, find def. {w°, w*, w?, w³, . ..} and then…

A:

Q: Show that if P(A;) = 1 for all i > 1, then P(N1 A¡) = 1.

A: We know that P(Ai) = 1 for i ≥1 Considering P(Ai), i=1,2,3,... to be independent events

Q: Given d, = and dy = Then d, and d, are not - conjugateif Non of these O D

A: Given d0 = -11  and  d1 = 56

Q: 1. Let n E N. Prove that n and nº have the same last digit.

A: See the attachment

Q: 6. Suppose (a, m) = 1. Prove that for any b, Σ an + b) 1 (т — 1). n<m

A: Given (a,m) = 1 We have to prove that for any b , ∑n≤man+bm = 12(m-1)

Q: 1. Let A = {z € Z:z = 2 (mod 5)}, and B = {y E N : y² < 90}. Compute the sets ANB, and B\ A in…

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Q: Show that if AB = I, then det(A) 0 and det(B) # 0.

A: Here given as AB=In find determinant both sides detAB=detIndetAB=1

Q: Let f₁=1,₂ = 1 and for all neN,n23,f=f+f2. Prove 1+√5 1-√√5 2 2 that for all neN, S √5

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Q: 3. Prove that if a > 1, then a"/n → o.

A: We used binomial inequality

Q: 1. Find the union and intersection of each of the following families: (i) M ={ n Z:nEN}, where n Z =…

A:

Q: 4. (a) Write down four distinct elements of the set {2 €C:2- E = 2i}.

A: Note:  We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: 2. Compute the convolution of the following signals for n 20 n n x1(n) = and x2 (n) = (;)" u(n) n…

A: Consider the expression of . Determine the z-transform of .

Q: 9. (Chapter 10, #20) Prove that for every n E N, (1+2+3+ · ..+n)2 = 13 + 23 + 33 + ...+ n³.

A: To prove the result ''∀n∈N,1+2+3+...+n2=13+23+33+...+n3'' by Mathematical induction: First prove for…

Q: 1. Suppose that an < bn for all n, an → L, and bn → M. Show that L < M.

A: As per our guideline we are supposed to answer only one question.kindly repost other question. The…

Q: 9. (a) Prove: If y>0, then there exists ne N such that n-1s y<n. ☆ (b) Prove that the n in part (a)…

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Q: Show that n is composite if and only if p(n) <n – Vn.

A:

Q: If n is not a poime then there exist Some ab€ 7h Such that ab # Itnk fox all

A: Definition: A number which can not be factored as product of two numbers other than the number…

Q: 2. Let A = and B = al. Show that det(A + B) = det A + det B if and only if a + d = 0.

A:

Q: Let a = (1 7 3)(5 94 2) and B= (2 3)(7 4)(5 1 8) be elements of S10. Then, a B is O (1 4)(2758 9 3)…

A:

Q: Let {0, 1}N = {(a, a1, a2, . ..) : an = 0 or 1, n E N}. • Construct a 1-1 correspondence between {0,…

A: let 0,1N=a0,a1,a2,....:an=0 or 1, n∈ℕ (i) construct a 1-1 correspondence between 0,1ℕ and all the…

Q: Show that if A is n × n with n odd and skew symmetric, then det(A) = 0.

A: Given:     A is n×n with n odd and skew symmetric.

Q: Let M = {14*: k E Z*} and A = {7k:k E Z*}. Then ACM A=M MC A None of these

A:

Q: (5a) Let A = {3, 4} be a subset of S = {1,2,... ,6} and let n E S. || n2(n+1)² Prove that if is…

A:

Q: Let n € Z. Prove that if 3 /n, then 3 | (2n² + 1).

A:

Q: 6.26. Let n > 2. Show that every element of S, can be written as a product of cranspositions of the…

A: Let n≥2. Show that every element of Sn can be written as the product oftransposition of the form 1,i…

Q: 12.42. Let a, be Z, where not both a and b are 0. Prove that if d = gcd(a, b), a = ajd and b = b1d,…

A:

Q: 1. Let S = {z E C| 26 = 1}. (a) For each element w in S, find def. {w°, w*, w², w³, . } and then |…

A:

Q: Which of the following are valid witnesses to show that 10n-1 is (n)? O C=1, k=100 O C=10, k=10 O…

A:

Q: Show that if A and B are both n × n, then (a) det(AT BT ) = det(A) det(BT ); (b) det(AT BT ) =…

A: Given A and B are n×n matrix.

Q: If X(z) = In(z - 2), then what's x(k O a. -ak O b. ak O c. -ak /k O d. others Oe. ak / k O O

A: Given that X(z) = ln(z - 2) We have to find the value of X(k).

Q: 6. Prove or provide a counter example for the following: (a) (AU B)nC = (AnC)U(BnC) (b) (An B)' = A'…

A:

Q: 21 Compute AB in the 3 0 4 Let A = and B -3 0 -2 -1 5 four different ways:

A: As per the instruction, we will be solving only the first part. Standard way,…