5. Let a, b, N E Z where N > 0. Prove that [ab] = [[a][b], where [r] denotes the remainder of r after division by N.

Q: Show that MnXn and Rn^2 are isomorphic.

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Q: Let X = {x E Z|x = Y = {y € Z]y = 3n for some integer n}. 6m for some integer m} and let %D a. Prove…

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Q: Calculate M, for f(x) = 6 · In(æ³) over [1, 2]. M6 ||

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Q: (b) En=1 m

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Q: Use Newlon's Me thod with the runclion $r(x)=x^{2}-2$ and inilial value $x_{0}=3: to calculate…

A: Givenf(x)=x2-2            and     x0 =0 newton method  xn+1=xn-f(xn)f'(xn)

Q: For any n > 1, prove that the irreducible factorization over Z ofxn-1 + xn-2 + . . . +x +1 is π…

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

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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…

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Q: 7. In ABCD, AC and BD divide each other in half at E. (a) Prove that ZDBC ZBDA. (b) Prove that ADBC…

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Q: Prove for all mEw and n E w, we have that m+ ·n=m・n+n.

A: Given that m,n∈ω. To prove: m+· n=m·n+n

Q: If n^2+6n+11 = x^2+y^2 for positive integer n, x and y prove [xy/2] is composite. Note: [t]…

A: Given :              n2+6n+11 = x2+y2              t ≤ t

Q: Let x, y, z, w ∈ Z and n be a positive integer. Suppose that x ≡n y and z ≡n w. Show that x + z ≡n y…

A: Suppose that x ≡n y and z ≡n w then, (i) we are assumed that  x-y=k1 n  and  z-w=k2 n  for some…

Q: Let n be a positive integer. The coefficient of x°y° in the expansion of (1+ xy+ y²)' equals

A:

Q: 3. Show that the function p : F → Ę which maps o to o is a bijection from F to Fe, where and ar to…

A: Let ϕ:F1→F2 be a function such that, ϕ0=0. Also given that ϕαr=β3r, where 0≤r≤6 When r=0,…

Q: 3.19 Suppose if r = 0, f(x) = {-1)" if r E n+1' n where n E N. Prove: f € R[0, 1], even though f has…

A: Sol

Q: Prove that E, k2* = (n – 1)2"+1 +2 for all n > 1. =D1

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Q: (b) Show that 1+ 2/2 has infinite order in R*.

A: Given that R=m+n2 | m,n∈ℤ (b) The objective is to show that 1+22 has infinite order in ℝ×

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

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Q: Find a number t such that In(4t + 3) = 5 %3D

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Q: 18. Let xo < x, < x2, let {yo, y, y2} be given. Find the polynomial q(x) with deg(q) < 2 and such…

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Q: Find the characteristic of Zm, × Zm, × · · × Zmr where m;'s are positive integers.

A: The ring ℤm1×ℤm2×...×ℤmk is a ring with unity 1,1,1,...k times Now, to find characteristic of…

Q: 3.2 Consider R2 with (, ·) defined for all x and y in R? as х and 2 (x, y) := x |1 =:A Is (:, :) an…

A: The objective of the question check for inner product.

Q: Prove that for every v = (x, y) E R? there exists a sequence v, = (xn, Yn) € R2, where rn, Yn are…

A:

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

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Q: 2. Prove that, for all k E N, 2 can be written as the sum of two squares.

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

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Q: prove for any 2 non negative integers x and y, where x>y, if their gcd IS Y, THEN THEIR LCM is x

A: The objective is to show that LCM(x,y)=x, given GCD(x,y)=y; where x and y are non-negative integers…

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

A:

Q: Prove that 223 a E lo,1) く 22 + 1영 ( 1-계 3(1-x2)

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Q: 3. Prove that Re(2-+z')<4 when z<1.

A: a.          x+y ≤x + y                                             (triangle inequality)b. Re(z1+z2)…

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: Let S = {n + m√2 : n, m ∈ Z}  To prove that the sum of any two elements of S is an element of S.…

A: Given: S=n+m√2:n,m∈Z  (a) Let a+b2∈S for all a,b∈Z and let c+d2∈S for all c,d∈Z Add both these…

Q: Prove the following in Z[i]: if a y and 3|y with a and B relatively prime then aß | y. in

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Q: If e denotes the unity element in an integral domain D, prove that me · ne = (mn)e for all т, п € Z

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Q: 15. prove that L. d= nd(n)/2

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Q: 1. Prove that fraction where m is z-4

A: Given: zn-1z-1 ⇒zn-1nz-1 ----(1) Let apply the limit in equation (1) limz→1zn-1nz-1 ----(2) Let n=pq…

Q: Prove or disprove: If (xn) and (yn) are subsequences of each other, then xn = yn.

A: Definition: Suppose ann∈ℕ be a sequence. A subsequence of this sequence is a sequence of the form…

Q: Supposernc N and n 4 0 Then show that rn is positive if and only if r is positive

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Q: If e denotes the unity element in an integral domain D, prove that me • ne = ( mn )e for all m, n ϵ…

A: We prove the identity as,

Q: prove that and = nd(n)/2

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Q: In ALMN, LN is extended through point N to point 0, mZMNO = (5x – 13)°- mZNLM = (x – 4)°, and m/LMN…

A: Solution-

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: (c) Prove that 10 is not separable.

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Q: If (an) has limit−1 and (bn) tends to infinity, does it follow that only finitely many terms of (an…

A: No it doesn't follow the statement

Q: Suppose n (a)= 7a + 4, determine a such that n (a) = 39. %3D

A: GIVEN n(a) = 7a+4 and n(a) = 39

Q: 3, 5x 60(x2 + *), 3 b) Y = In 7.

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Q: By expanding both sides of the identity (1+ x)"+4 = (1+x)"(1+x)* prove that ("**) - (") + «(,",) +…

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Q: 4. Verify thatf (x) = x³ – x2 – 20x + 6 satisfies the three conditions of Rolle's Theorem on [0,5].…

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Q: Let a E Z, a #0. Prove that for any positive integers m, n with n<m, a2" +1 divides a2 -1. a) a even…

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

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.