4. Prove: Vn e Z, if n>2 then n -1 is composite. 5. Prove: Va,b e Z, if a|b and a|c then a|(2b – 3c)

Q: 9. Suppose that Y CZ and that n(Y) = a and n(Z) = b (so a and b are numbers!). Find n(Y N Z), n(Y U…

A: Note: Hi there! Thank you for posting the question. As you have posted multiple question, as per the…

Q: Let x, y, z e Z Prove that if x – 2y + 3z is odd, then 4xy – 6z is even.

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Q: Let x, y, z E R. Prove that if x – 2y + 3z is odd, then 4xy – 6z is even.

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Q: Show that for n, m e Z+, gcd(2" – 1,2 – 1) = 28cd(n,m) – 1.

A: To prove : gcd(2n-1, 2m-1)=2gcd(n,m)-1where n,m ∈ℤ+

Q: 5. Show that if x,y∈Z and 3|x^2+y^2 then          3|x or 3|y 6. Show that if c=sa+tb where c=(a,b),…

A: Please check that x=1,y=3, then x^2+y^2=10 does not divide by 3. So  must 3|x and 3|y instead of 3|x…

Q: 4. (а) Let a, b, c E Z. Prove that, if a | b, then ac | bc, but that the converse may be false. (b)…

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Q: * If Iz| = Re(z), then z is

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Q: 3 2 If A = use the Cayley-Hamilton theorem to determine A?. 5 1

A: We can solve this by using given information

Q: 2. fow く2 <a 30 20 Prove thot 大2 %3D n 21

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Q: 2. Suppose A is an m x n matrir, x ɛ Rª and Ax = b. Then b is in 1) R" | 2) R™ 3) RmfnT 4) R™n

A: Given :  A is an mxn matrix , x∈Rn and Ax=b   To find :  b is in which space of the given spaces.

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: Suppose gcd(a, b) = 1. Prove that if r|a and s|b, then gcd(r, s) = 1

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Q: (3. Suppose T is a linear tronstormation from V to w and that {š X is a LI. söt of ve ctors in V.…

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Q: 8 ) ) Let A=[0 ;], B=[; %], C-[; !] C=[o a) A &B& C are nilpotent in M₂(R) b) A &B are nilpotent in…

A: Nilpotent element: An element x of a ring R is called a nilpotent element if there exists smallest…

Q: In the given figure, suppose M2TUV=(4x-8)° and mzTSV= (x+16)°. Show that mzTSV is hal of M TUV. * S…

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Q: 1. Prove the following in SDE. P+ Q, Q→ R, R→ S - P –→ (R A S)

A: When we infer a conclusion from a given premises in order to produce an argument, we use laws of…

Q: $ = g(ø, v, ø) = aø + - b = g(4, 4, 4) = arb + which can be rewritten as ayoy – ado? + Bo², ayoy –…

A: Given: ϕ=gϕ, ψ, ϕ=αϕ+βϕ2γψ-δϕ and ψ=gψ, ϕ, ψ=αψ+βψ2γϕ-δψ To show: δ1-α+βϕ2-ψ2=0

Q: Hi, parts a) and b) of this question are attached.    Thanks so much!!!

A: For (a),

Q: 10.) PN= ON: N MN- LV=U6

A: As per our guidelines, we are allowed to answer only one question. So I'm providing answer to only…

Q: Let a, b, c E Z. Prove that if a b and alc, then Vk EZ we have a (b+ ck).

A: ab  &amp;  ac   Since ab, therefore b=a·m   ,  where m∈ℤ Since ac, therefore b=a·n   ,  where n∈ℤ

Q: 5. Prove that if A and B are similar then dim(ker(A)) = dim(ker(B)).

A: We have to prove dim(ker(A)) = dim(ker(B)).

Q: 3. Suppose AB=RS and mzB = mzS. What other information is needed to prove that AABC = ARST by SAS?…

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Q: ces. 1 W= d b such that Z = ax + bY. calars a and b such that w = ax + bY? = 0, is it true that a =…

A: We have to find the following conditions according to the given information

Q: Let m eN and a, b e Z be such that (a, m) = 1 and (b, m) Let h = 0m(a), k = 0m(b), and let n = 0m…

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Q: Is S = {a + b√(2) : a, b ∈ ℤ} dense in ℝ?

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Q: 5. Let H= [ (û ¿)I a,b,c EIRI, show that: (H,+) is a sub gromp of (M,+).

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Q: Suppose that A ¥BSCN D and x E A. Use the method of "proof by contradiction" to show that if x e C,…

A: Given data : x∈Aandx∉C To show : x∈B

Q: 4 Let X,) be a pankizlly oadered e For x,y € X deftre xLy eff x sy and x#y. Pucose the Fallowiag:…

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Q: Prove by contradiction that if m, n € Z, then m² - n is even or m-n-1 is even.

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Q: Show that for n, m e Z+, gcd(2" – 1,2m – 1) = 2gcd(n,m) – 1.

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Q: 3. Suppose a, b, c E Z. Prove that if a? + b = c²,, then a is even or b is even.

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Q: Assume that =y+In(x), and n=x, then u xX а. O a. 1 Ob. 1 1 с. O c. x² d. 1 Unn x 1 O e. x² е. 2 +Unn…

A: ANSWER :

Q: In (R, Tµ), if A = (1,3) n (Q – Z) then A = - %3D O a. [1,3] n (Q- Z) O b. R O c. [1,3] – {2} O d.…

A: Given that, In ℝ,ζυ, if A=1,3∩ℚ-ℤthen calculate the value of A.

Q: 9. Prove or disprove. (a) Vn e Z*, n² < 2". (b) Vm e Z, if m is odd, then m² is odd.

A: In this question we have prove if this equation are valid or not.

Q: 1. For n 2 2, let A = [a E M,(C). Suppose A is diagonally dominant. Show that if af # 0 for all i,…

A: We will use the basic knowledge of matrices and diagonal dominance to answer this question in…

Q: Subst ), we nave > Zn-h 1 kn+1 h=1 = ) kin-1 Vn | 2 EWn-p 1 = Vn-1. kn p=1 Vn+1 Wn - H Hence, we see…

A: given equation (5) wn+1-μ∑p=01wn-p=∑h=12zn-hzn-ε  and  zn+1-ε∑h=01zn-h=∑p=12wn-pwn-μ also equation…

Q: 4.2. Let a, b E Z, where a 0 and b 0. Prove that if alb and bla, then a = b or a = -b. -b.

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Q: .....………., ɑn are in A.P., where a; > 0 for all i, show that : 1 If a1, a2, a3, 1 1 n- 1 %3D Va1 +…

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Q: 4. A region D in RN is said to be convex if and only if for every u, v E D and t e (0, 1), then tu +…

A: Given: A region D in ℝN is said to be convex if and only if for every u,v∈D and t∈0,1, tu+1-tv∈D.…

Q: 2) Let mf lu and (a,m)=| a,bE Z such that Let h= Om ( a) show and and (h,m)=1, %3D let. ne OmCab)…

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Q: Let X1,., X, N(u, o?). iid N(H, o?). Find E(), if ju = X1 + (X2 + + X(n %3D 2(n-3) 17 16 4 O 17(n-2)…

A: Given that, E(X1)= E(X2)= ......E(Xn)= mu

Q: Prove: Vn e Z,if n is odd, then 3n+5 is even.

A: Consider the provided question, To prove: if n is odd integer, then 3n + 5 is even. Proof by…

Q: (c) If A E M,(R) is non singular and n 2 2, prove that det(adjA) = (det A)"- (d) If A E M,(R)…

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Q: 5|8" – 3", Vn EN

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Q: By using Caushy Rieman prove that (2) and (3) (2) 4ca) = e (3) .w prove Thot()-

A: (a)

Q: 9. Given: AMNP - AQRS, mZM = 56", mZR = 82°, MN = 9, QR = 6, RS = 7, MP = 12 Find: a) mZN c) NP b)…

A: Explained below

Q: Q1: Prove that ; If CE AB, D E AB and C# D, then CD = 2 %3D

A: Key point: 1. There is only one line possible from two distinct points.2. The line is expressed in…

Q: Let X,Y⊆UX,Y⊆U. Suppose that n(X)=26n(X)=26, n(Y)=24n(Y)=24, and n(X∩Y)=20n(X∩Y)=20. How many…

A: To derive the number of elements in the set X' union with Y , from the given data

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

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Q: Given Jao ao = = 9, a₁ = 36 a1 an+2 = 7an+1 Show that Vn N, n ≥ 1 10an, 20 an=3.2 +6.5n

A: