For integers m, n, p, q E Z, n, q0, use this and the field axioms mq+ng and (a) Prove + 2 mp = n 11 nq nq (b) Show that the field axioms also hold for rational numbers Q. (c) Prove < mq> np for n<0,q> 0. 71 9

Q: Prove that (-1+ iE)" + (-1- i5" has either the velue 2n+! or the value -2" f ロ ww. š any integer…

A: The cube root of unity is written as:

Q: Which of the following sets is not closed for the operation °, which is defined by the rule a°b = a…

A: The objective of this question is to determine which set of integers is not closed under the…

Q: Let a = 3/2 + 34. (a) Compute a³ and write it in the form pa + q where p and q are integers. Use…

A:

Q: Let R[r] be the set of all expressions 00 a = do+ajx+azx += i=0 where a¡ E R for all nonnegative…

A: We will be solving part (b) as mentioned. (b). Given ℝx be set of all expressions…

Q: Let x be a real number. Which of the following is the same as P(Z x) c) P(Z ≤ -x) d. none of the…

A:

Q: Prove that the statement is true for every positive integer n.

A: -11+-12+-13+---+-1n=-1n-12. We prove this statement by using the Principle of Mathematical…

Q: 7. Prove that 4 3m² – n² for all odd integers m and n.

A:

Q: 4) Solve the following linear system or indicate the nonexistence of solution. X1 – 2x2 + 5x3 – 3x4…

A: Or, take x2 = k Hence, solution of the system is x1 = 1+2k, x2 = k, x3 = 1, x4 = 2

Q: The principal represents an amount of money deposited in a savings account subject to compound…

A:

Q: 2. Show that for any real number c, there is exactly one integer in the interval (c,c + 1].

A: As per guidelines, I can do only first question.

Q: Which of the following are continuous and piecewise polynomials with respect to the nodes o, x1, x2,…

A: Continuous and pocewise continuous of the polynomial

Q: If A = {r|- 23 < x < 3how many negative integers does Acontain? Note that: 'I' means 'Such that'

A:

Q: Exercise 9. We can find a non-constant polynomial that is bounded over the set of complex numbers.…

A: This is false

Q: Let a and b be positive integers and let b= qa +r, where 0 ≤r <a. Prove that (a) 2b - 1 = 2" - 1…

A: Let  and  be two positive integers.Let  where (a) We  have to prove that      Now, observe that…

Q: (4b) Prove that there exists no positive integer x such that 2x < x² < 3x.

A:

Q: Show that the polynomial 1+x+x² +...+xP-1 . where p is a prime number, is irreducible over the field…

A:

Q: For each integer n ≥ 1, let Q be the set of all n-tuples of rational numbers. For example, Q³ ; a,…

A:

Q: Prove that if x is a rational number andy is an irrational number, then x + y is an irrational…

A: As you have asked the problem ( b) only. I have solved this problem in two steps by showing the…

Q: Find the smallest positive integer in the given set. (a) {6u + 15v : u, v ∈Z}    (b) {12r + 17s : r,…

A: Given: (a) { 6u+15v:u,v∈Z }(b) { 12r+17s:r,s∈Z }

Q: Using Exercise 2, show that the set of rational numbers Q forms a subfield of the real numbers R.…

A: A subset K of a field F is a subfield if K is itself a field. We know that the set of real numbers…

Q: Prove or disprove that there exist rational numbers x, y EQ such that √11: = x+y√5.

A: Prove or disprove that there exist rational numbers x, y  in Q such that √11 = x+y√5.

Q: For the following proof, determine which of the statements given below is being proved. Proof.…

A: Contrapositive: The contrapositive of the statement, " If P, then Q" is " If not Q, then not P".

Q: 5. Let F= [3,2,1] and r= [2,1,–3]. Compute the following: %3D a) Proj „F. b) Orth rF = F – Proj „F.…

A: We know that  ProjrF=r.F∥r∥2r Now, r.F=23+12-31=5 ∥r∥2=r,r        =2,1,-3,2,1,-3        =22+11-3-3…

Q: Evaluate the following commutators [Â, B] = ÂB – BÂ. Hint: use an arbitrary function that cancels at…

A: a x,PxPx=-ih∂∂xx,Pxφ=x,-ih∂∂xφ=-ihx∂φ∂x+ih∂∂xxφ=-ihx∂φ∂x+ih∂φ∂x+ihφ=ihφ∴x,Px=ih

Q: Let L be the infimum and let U be the supremum of the set {x≤R : 2x² - 9x − 31 < 0} Find the sum…

A: Given set { x∈ℝ  :  2x2-9x-31&lt;0

Q: Prove that the roots of X2+(1-K)x+k-3=0 are real for all values of k?

A:

Q: How would you use the existence and uniqueness theorem here?

A: Here the given initial value problem is,.Existence and Uniqueness theorem :The solution of an…

Q: 20. (a) Fill in the blank: 1+3+5+..+ (2n – 1) = k=1 (b) Use part (a) to prove that the sum of the…

A: This is an arithmetic series with common difference (3-1)=2 And this has n terms in it So we use the…

Q: Let be the following set of rational numbers: Consider the algebraic structure (R2, +, ·), where +…

A: Field :-  A commutative ring F with unity is called field, if each nonzero elements of F has…

Q: Find the term independent of of in the expansi Ansion of √x 3 + 3 222 to

A: Given expression is x3+32x210. The aim is to find the term independent in the expansion of the above…

Q: a) If q(n) = byn* + Bk-1n*-1+...+ b1n+ bo is a polynomial in the variable ne N with be # 0, show…

A:

Q: Euler's formula for generating amicable pairs of integers is the following rule: = Suppose n> m > 1…

A: Note : In the Euler's formula for generating amicable pairs it is r = 2n+m (2n-m + 1)2 - 1 and not…

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: Find the length of (3et − 3, 4et + 7) for 0 ≤ t ≤ 1

A: To find the length

Q: Show that the polynomial 1+x+x² +... +xP-1 , where p is a prime number, is irreducible over the…

A:

Q: (a) Prove that the function f is well-defined; that is, the value f(q) of the function at a rational…

A: Given the notation , and using this defining the set . Then the function  defined as:.It is well…

Q: Prove that for all real numbers Ex)y= -(1u)

A: We have to prove that for all real numbers (- x)y= - (xy)

Q: 1 f X~ N(0,0²), then X² ~G (¹2¹2) Prove this statement

A: Given  X~N(0, σ2)