(c) List all elements of A that are related to 5. (5 S x) . This is the class of 5 denoted by [5] .

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: Suppose that j and k are positive integers such that j< k. Give a combinatorial proof of t identity:…

A:

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: • x² – 3 = 0 x2 – 3x = 0 x² + 4x = -4

A: NOTE: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question…

Q: Find the value of k such that A is singular. 1 0 4 5 -1 0 3 3 k k =

A: Topic:- Matrix

Q: 7. Consider an element z = the order of r is the least common multiple of the orders of its…

A:

Q: 1. For every element x in A, there is an element y in B such that (x, y) E F.

A:

Q: Suppose that Φ is a homomorphism from D12 onto D3. What isΦ(R180)?

A:

Q: 4. Show that in the following sense: for any ɛ > 0, there exists Ro >0 so that for any R> Ro, R rp

A:

Q: Which of the following is homogenous of smallest degree? (A) In x - In y B x+y x-y C √x-y x² + y4 X…

A:

Q: Prove that S4 is not isomorphic to D12.

A:

Q: relation R on Z defined

A: Given, a relation R on Z defined by mRn ⇔ 7|(m − n) for all m, n ∈ Z.

Q: Expand f(z) =1/z2  in powers of z+1

A:

Q: 5. Consider the functions l : N → Z by if n is even l(n): 1- n if n is odd 2 Is l one-to-one? onto?…

A:

Q: For the K class classification problem, prove that f(x) = argmax-1,.K K) is the Bayes classifier.…

A:

Q: Let a be a non-zero element of Z43. Explain what it means for |a] to be of order w, and hence prove…

A:

Q: 1 Expand in ascending powers of x, up to and including the term in x', simplifying the V(1 –2x)…

A: To solve the following problem

Q: Q2/(A): Let L.N are submodules of M and n:M - M/N be the natural R-homomophism. Is n(L) submodule of…

A: Given that,                 L and N are submodules of M .  Also, π : M → MN is a natural R-…

Q: Determine discrete numeric function corresponding to the following generating function: A(Z)=1/1-Z³

A: Given generating function is A(z)=11-z3 We can write it as A(z)=1-z3-1⇒A(z)=1+z3+z6+z9+....

Q: Compute |A| for the following:

A: We have to compute A for the following given matrix: A=210121012 We know to determine the…

Q: Prove that d24 and s4 are not isomorphic. List elements and there order.

A:

Q: Given that the points (4, 5), (5, 6), and (6,8) function f, find f1(6). points on the graph of an…

A: Given that f is invertible Function ⇒f and f-1 are one to one and onto

Q: 1. For every element x in A, there is an element y in B such that (x, y)EF

A: Remark: Let A and B be two non-empty sets. A relation R from A into B is subset of A*B. A function f…

Q: 12. Start with the identities x2 1 P, (x) = xPn-1(x) + *P(x) n P, (x) = xP-1(x) + nP,-1(x ) (a) Show…

A:

Q: Show that x2 + x + 4 is irreducible over Z11.

A:

Q: Show that x2+x+4 is irreducible over Z11.

A:

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

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

Q: Is z(√2) isomorphic to z(√3)?

A:

Q: Which of the following is homogenous of smallest degree? A In x - In y x-y B /x-y x4+y* x+y

A: Let's find.

Q: Show that Z4 is not isomorphic to U(12) by comparing the orders of elements.

A: z4=0,1,2,3 is group w.r to +order of elementsoa=n⇒an=e where n is least positive…

Q: Determine the z transforms of the following sequences 3t (a) {(-} )'} (b) e

A:

Q: 4. Prove C(n, r) = C(n- 1,r- 1) +C(n- 1, r) %3D using a combinatorial proof.

A: We know, C(n,r) = n!/((n-r)!r!)

Q: - [: ]} а b Identify all of the nonempty submatrices of A d

A: →A matrix obtained by deleting some of the rows and/or columns of a matrix is said to be submatrices…

Q: (b) There exists LE R such that L4 Size(O(n)).

A: Given below clear explanation

Q: Find the smallest integer in the given set a.{ xϵ Z f x &gt; 0 and x = 4s + 6t for some s, t in Z }

A: Consider the equation: x=4s+6t……1   where: x, s, t∈ℤ and x&gt;0

Q: m Show that E (")C")= ("*"). Using Combinatorial Proof. \m+p. k=0

A: We will use equating coefficent method

Q: To which of the following does the Well-Ordering Principle directly apply? x(x + 1) x E Z 2 z*} A. O…

A:

Q: p be an irreducible reprsentation of G

A:

Q: Represent algebraically and graphically the set of values of z for which |z+3|=|z−2i|

A:

Q: Let dg denote the number of derangements of order k. (a) Prove the identity C) d = n! with a…

A: Given dk is number of derangement of order k. We know that for a set 1,2,…,k, the derangement dk are…

Q: Compute the convolution x[n]*h[n] for all n> 0, by using Graphical or Tabular method. x[0] = 3, x[1]…

A:

Q: Define p: Z16 Z16 by p(x) = 4x is a homomorphism then Kerø = a) {0,12} b) {0,4,8,12} c) {4,8,12} d)…

A: Given  ϕ:ℤ16→ℤ16  by ϕ(x)=4x is a homomorphism. To find: Ker ϕ

Q: 6. If X has full rank, so that P = X(X'X)-1X', prove that C(P) = C(X).

A: The rank of a matrix denotes the number of independent rows or the number of independent columns in…

Q: Given a relation R on Z defined by mRn 3|(m – n) for all m, n E Z. Find the equivalent class…

A:

Q: Determine the z transforms of the following sequences 3t (a) {(-} ) } (b) е e

A: Given

Q: Let n E Z. Show that f(x) = x3 – nx +2 is irreducible over Q for %3D n + -1,3, 5.

A: We shall find all n ∈ Z such that f(x) is reducible. Assume that f(x) is reducible. As f(x) is…

Q: . In Z[V-5], the element 29 is: None of the choices Irreducible O Unit O reducible

A:

Q: Find infinitely many polynomials f(x) in Z3[x] such that f(a) = 0 forall a in Z3.

A:

Q: In Z[V-5], the element 29 is: None of the choices Irreducible Unit reducible

A:

Q: Find a generating function for a,, the number of partitions of r into (a) Even integers (b) Distinct…

A: a. The polynomial ( 1 + x^2 + x^4 + ...) represents the number of 2's in our partition. The…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.