7. Let c t- are both even or both odd. Let that o and(t 0, t be permutations in S. Prove t

Q: For two R-modules Mand N Show that Hom (M,N) any with the composition is not agroup?

A: Given that M and N are two R-modules. We have to show that HomM,N is not a group under composition.…

Q: Let n, k, m EN with n2k21. prove that

A:

Q: If k >= 2 and x in R^ k prove that there exists y in R^ k such that y ne0 but xy = 0 Is this also…

A: If x has any components equal to 0, then y can be taken to have the corresponding components. equal…

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

A: Use the following definitions, to obtain the solution. The Euler phi function for the positive…

Q: Let C: z-3 and I – 6dz. ONone of these

A:

Q: 1) The r ing of in tegers (Z) is not ideal of Yeal umber (R) n O True Fulse

A:

Q: 1. Prove that the followingn is a theorem in SD. Av - A

A: Theorem :-

Q: 5. Prove that (Z, × Z6)/((2,3)) S Z12.

A:

Q: Suppose that > ak = a and let c E R. Prove, in all the that (c-an) = gory detail necessary, = c•Q.…

A: We will give two proofs one based on partial sums and the other will be a direct proof.

Q: Let S = {10n – 1: n e Z}, T be the set of odd integers, and U Prove that S CTNU. {5т + 4: т E Z}.…

A: Consider the given information. Here, S=10n-1:n∈Z And, U=5m+4:m∈Z And, the set T is the set of odd…

Q: 2. Let A € M₂ (R) be given. (a) If AA¹ = 0, prove that A = On. Does this hold if A € M₁ (C)? (b)…

A: This is a problem of Linear Algebra.

Q: Let a and ß belong to S,. Prove that a-'ß-laß is an even permutation. n°

A:

Q: 5. Let r € R and A, = {q € Q : q < x} . Prove sup A, = x.

A: According to the given information, Consider the given set Ax:

Q: Let a e Z. Prove that if 6 | a and 10 | a, then 15 | a.

A:

Q: Let (W, t > 0) be standard BM. Show that P(n|w,1 – Wa sc) → as n → 0.

A: From the given information,  (Wt, t≥0) be standard Brownian Motion. Note: In the given situation,…

Q: 1. Prove that if A is infinite and a E A, then A – {a} is also infinite.

A:

Q: Show that there exist m and k such that ma + kn = 1if gcd(a, n) = 1

A: Let gcda,n=1.

Q: (c) Show that a permutation o E S, commutes with r if and only if the permu- tation f(o) commutes…

A:

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: For allne z, gcdin,nt)=l.

A:

Q: Let a and b belong to Sn. Prove that ßαß-1 and a are both even orboth odd.

A: Let, a and b belong to Sn.   Consider α is the product of a 2-cycles, and β is the product of b…

Q: 1. Prove that if A is infinite and a e A, then A – {a} is also - infinite.

A:

Q: 11. Let S= {| -2 }. Show that u= E span S. 1 2

A:

Q: Prove that if n 2 3, the only element o of S, satisfying oy =ya for all y E S, is the identity…

A: Let n≥3. Let σ∈Sn such that σγ=γσ for all γ∈Sn. Suppose σ≠i, the identity Then there exists…

Q: Suppose T: R + P, has an action on a

A: Given, And

Q: Let Sn=Vn. Find an and show that an O but that Sn=ER=1@k diverges. ->

A: Solution:-

Q: 7. Prove: Vn e Z, n is odd if and only if n is odd.

A:

Q: discrete

A: In method of contradiction we start by assuming that given statement is false. Then prove that it is…

Q: 5. Let z = a + bi and z2 = az + bzi. (a) Find Im[z,z+()?-z,z ]. (b) Prove that (7)² = z,?.

A:

Q: Let x, y, A, u E Z. Prove that if 1 = Ax + µy then r, y are relatively prime.

A:

Q: 2) Let a,bE Q such that a<6. Prove that ax <bx.

A: The statements can be proved by analyzing the results from the given statements. The other methods…

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

A:

Q: Prove that the set S = {n² : n E N} is denumerable.

A:

Q: Prove or disprove the following: Vne Z, n³ + n²+1 is odd

A:

Q: Ands 1. Let m*(A) = 0. Prove that for every B, m*(BUA) = m* (B).

A:

Q: @ 2 show Hhat fox) = e*-x3+2 2+2 is a %3D Permutation on R.

A: In the give question we have to show that the function f(x)=e-x-x3+2 is a permutation on ℝ.…

Q: (4) Let A CRbe a et ith greatet element a E A Prove that sup(A)=a

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

Q: Let z = x + yi, z2 = 1– 0.5i , and z, + z2 = 3 – 1.5i. Find In(z,) – In(z2) %3D A) In(25) ± 2niN D)…

A:

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: 2. Suppose that {v1,... , Vp} is an orthonormal set in R", and that x E R". What can you say about…

A:

Q: 2) Suppose S,T E L(V) are such that ST = TS. Prove that null S is invariant under T.

A:

Q: let A to have0 al an must by

A:

Q: 3. Given any nonempty set A equipp +: A x A → A, for any two nonen such that J is a permutation of I…

A:

Q: 5. Let TE B(H). Show that (ker T)- = ran T.

A: Given: T∈BH and T† is an adjoint of an operator T. To show: ker T⊥=ranT†

Q: Prove that if c=12n+4 where n in Z then c is not a linear combination of 27 and 18

A:

Q: 7) Show that K± = det [ T 4 J.

A: Sol

Q: 12. Prove the following (A+ BB")-B= AB(I+ BT AB)-1 by express the B+BB' AB in two forms. O Type here…

A:

Q: 4) Let x, y, z E Zg. Prove or disprove the following: Ifxy = xz, then y = z.

A: