questions Let F be a f.g free R-module- Verily tieat Hom (F, R) R or not. (R is^commutatine is a free Romodele

Q: Can you prove that it is injective by contrapositive? Also, how is f surjective when y exists in R?

A: Different equivalent definitions can be given for injectivity. The function f:A→B is injective if…

Q: Find Souution of ODE xy"+y't x²y 20

A: Given, the differential equation                  xy''+y'+x2y=0                 ...(1)

Q: Suestane what be Scrd abaut the senes? con from the limid Comporson test 2) le 6) None we convege c)…

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Q: Then slouw that E Alasf bea, Couly, seguene in D. is also Cauchy.

A: Here uniformly continuous is necessary condition. Here is proof.

Q: If sup(S) exists, then it is unique ylgn ihi

A: The solution is given as

Q: on Give an crample of a dis continous bijection fo[at] = [1,2]

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Q: 2. Answer the questions with Yes or No, and then explain why yes or why not. Justify your answer. a)…

A: The solution of given problem is given as follows :

Q: B₁ = {(a, b): a, b are realt B₂B RIB is finite} I, is sopology generated by J₂ is Sopology generated…

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Q: L Idempotent Law 2 Double negation law b.) p (qVr) E-pV (qVr) 3. Associative law c.) (pv q)A [~…

A: From the given we have to match pair of law and the logical statement.

Q: <x tet fecl [a,b] and oege have we = f(n) ganze x Je count + fit-LED Plant at y - (x-[x]) f(x) +…

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Q: If it is possible to label each element of an infinite set S with a finite string of keyboard…

A: As per the guideline, I am supposed to solve only first question from the given problems.Given that…

Q: al let (xt) betopology space and BCY then B is closed in y iff there is closed set CinX st B-COY

A: Let (X,Γ) be topology space and B⊂Y To Prove: B is closed in Y iff there is a closed set C in X s.t…

Q: Find, the domain of each of these, ped pred cates where the doman is the set of untegers లు రుంకు

A: Disclaimer: Since you have posted a question with multiple sub-parts, we will solve first three…

Q: Find oxthogonal uncdary diagonalizing an or matri'y to lMowing tor One 2 (a) (b) 3 -2 -2

A: A square matrix A is called as orthogonal matrix if AxAT=ATxA=I where AT transpose of matrix A. and…

Q: A subset (a,b) ≤R is T2-space in (R,Tu) * 1 true 2 False

A: Given: A subset (a, b)⊆ℝ is T2-space in ℝ, Tu. To determine: Whether given statement is true or…

Q: Solution (a) R = {(a,b) = (Z>0)²: Ak € Z such that a o. There can exist no real number c such that a…

A: The solution to part (a) looks correct, although it would be clearer if it explicitly stated that R…

Q: Let f be the map sending each Comp lex num bey z> Show f(2,Z2) = f (2,) f (2») %3D

A: The detailed solution is as follows below :

Q: (O Show that AdditPon closed i's a binasy binazy operation of the Set sver Real num beð and Complex…

A: The objective is to show that addition is a closed binary operation over the set of real number and…

Q: 1 If MR =: 1 lo 1 1., then the relation ol 1 Ris Reflexive Transitive Anti-symmetric Symmetric

A: To find: The type of relation R. Concept used: Reflexive: If all the diagonal element of MR is 1,…

Q: P(1,1, 1) Q(-1,1,1) R (0,0,4) Datcrmine an cquation of the planc II in IR that contains P.Q and R

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Q: Question 2 Given is set A = {1,2,3,4}and relation RCAXA, define on the picture 2 This realation is:…

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Q: 12) The value of f, g,f,and g! are given in the table below live f (x) 3. -3 3 6 -2 -1 2. A) Eind h…

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Q: Shows that the pair are not equivalent in FOL.  ¬∀x∀yB(x,y), ¬∃x∃yB(x,y) Show through specifying a…

A: To show that the given pair of formulas are not equivalent in First-Order Logic (FOL), we need to…

Q: Usina rules of differentadion to find the detvative Of the functian メ-5ゃーメ %3D It eX

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Q: and n 12/22 -42 Z+6z Z722+2 2ナ z2, 4z+3

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Q: OShow that the set (1,2] is not compact 2Show that the set [1,2) is not compact 3Show that the set…

A: 1.To show that (1,2] is not compact

Q: Prove or disprove the following statement: A) Every T'3-space is T4-space?( explain your answer)

A: The statement "Every T3-space is a T4-space" is true. To understand why, let's first define the…

Q: a) Sean A, B, C sets and f: A → B, g: B → Cinvertible functions. Prove yes gof is invertible, so for…

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Q: on 1: a) Let O be the set of odd numbers and O' = {1,5, 9, 13, 17, ...} be its subset. Define the…

A: Since the part b is independent of the first part so I am answering the first question only for the…

Q: Find the transitive closure of R if MR is

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Q: Q) let P aprobabiity funckion on S: La,a,a) Find plai) If:- i) pcaz) = and plas) = ii) pc{ãa, az} ,…

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Q: La) Emplain why is entive ? the function fcz) = 5z"-6+zcosz -ze

A: z=x+iy⇒5z4=5x+iy4=5(x4-6x2y2+y4)+5i(4x3y-4xy3) ---&gt;eqn(1)⇒xcosz=x cos(x+iy)=x(cosxcos hy-i sin x…

Q: A Moving to the next question prevents changes to this answer. Question 2 Let AcR be bounded and set…

A: Choose the correct

Q: About the universal quantity (see picture)   and A, B, C are quantities in this universe that…

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Q: Please poove 18 ŏ bound of (३5) fश्ठ also proove ppes an sup13,5)%35

A: (.) Upper bound:    "A number that is greater than or equal to any number in a non-empty subset of ℝ…

Q: ७) Let యర with Aand be commutetive Xing that that 1of iniby Then Such Poove ২ A.B AnB

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Q: tet U and I are usual and cofinte tel toplogy on R show that I and U is counnected. or not?

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Q: 1. Evaluate lim dt (show all work and logic)

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