Algebra Consider the set (a, b) x (c, d) in R² , where (a, b)x(c, d) := {(x1, x2) E R² : a < x₁ < b, c < x2

Q: Consider V = {(2,y) : x, y ≤ R}. Let and ⋆ be operations on this set defined in the following way:…

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Q: Let F= R and V = R*. Let W, = {(x1,x2,X3,X4) € R* : x1 – x2 + X3 – x4 = 0 and x1 +x2 = 0}, W2 =…

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Q: Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (х, у)…

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Q: [Falkner Section 11 Exercise 26] Let A, B, and C be sets. Prove that if f: A → B and g: B→ C are…

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Q: а. v(Ã ·) = Ã × (V x B) + B x (v x Ä) + (Ã ·v)Ē + (B · V)Ā b. · (Ã × B) = B · (v × Ã) – Ã · (V × B)…

A: Part(a): ∇A⇀·B⇀=A⇀×∇×B⇀+B⇀×∇×A⇀+A⇀·∇B⇀+B⇀·∇A⇀ From the left-hand side of the given equation,…

Q: i: Let X and Y be topological spaces. Show that X Y X -Y but X -Y X - Y.

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Q: Find u, u = (-1, 4), v = (4, 5) (a) ||u|| v, uv, and d(u, v). (b) ||v|| (c) (d) U V d(u, v)

A: The given vectors u=-1, 4, v=4, 5. We have to find the values of u, v, u.v, du, v.

Q: Let Ai = {i, i + 1, i + 2, . . .} find ∩∞(infitity)i=1Ai and ∪∞(infinity)=1Ai Explain.

A: We have to find the intersection and Union of given set.

Q: Show that V spans P² where V = {1 + 2t + 2t², t + t², 1 − t²}

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Q: A) Let f: Z→ N as follows: 2n – 1 for n > 0, - f(n) = -2n for n < 0. Show that f is a bijection.…

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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: assume that Pr[A]=0.25Pr[A]=0.25, Pr[B]=0.35Pr[B]=0.35, and Pr[A∩B]=0.1Pr[A∩B]=0.1. 1)Pr[A|B] =…

A: Given: P(A) =0.25 P(B)=0.35 P(A∩B) =0.1

Q: Let u, ER" prove that ||ū + ï|| ≤ ||ū|| + ||v||

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Q: a-c

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Q: Let ƒ : Z → Z × Z be defined as f(n) = (2n, n + 3). Select which statements are true. a) for any in…

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Q: a) f(AUB) = f(A) U ƒ(B). b) (i) f(An B) ≤ f(A)n ƒ(B). (ii) Find a function f {1,2} → {a}, and A, B C…

A: a)  We  have  to prove  that  Now,      Thus,  (Proved)[since, b) i) We  have  to show  that Let  be…

Q: 1. Let Q(v2) = {a+b/2| a, b E Q}. Prove that Q(/2) = Q[x]/(x² – 2).

A: We have to show that isomorphism means there is function ϕ such that we have to show that it is an…

Q: The domain of f(x) = V x2 + x – 6 | is : a) ]-4, 1[ b) ]-00, 4[U]1, +0[ c) ]-00,-3[U]2, +00이 d) ]-3,…

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Q: 1. Prove that for all sets X, Y , if XU Y = Xn Y, then X = Y. %3D

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Q: Write an equational-style proof for the following theorem schema (r→ (pVg)) + ((rv (pvq)) → (p \q))

A: Write down the given statement and the goalGiven:Goal:

Q: QUESTION 5 Solve the given ordinary differential equation by separation of variables. dy = 2y dx…

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Q: 3. Write the cov(a + bx, c + dY) in terms of var(X), var(Y), and cov(Y,X) (Math proof)

A: Using the properties of covariance, the expression simplifies to:cov(a + bX, c + dY) = ac * cov(X,…

Q: 1. L: V3 V4 defined by L(a, b, c) = (a - b,b+c-d, a+b-2d, b-c-d) 2. L: V4 → V3 defined by L(a, b, c,…

A: According to Bartleby guidelines we can only solve 1 question or three subparts. So put other one…

Q: |(u+ v) x (u – v)I = 2|u x v|

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Q: 9. Let D be an integral domain such that (x + 1)² = x² + 1 in D[x]. Char D = а) 0 b) 1 c) 2 d) 3 O…

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Q: Is the set {(-3,0,3),(1,-1,2)(4,-2,2), (2,1,3)} of R' linear dependent over R?

A: Clearly dimension of R3 is equal to 3. So at max. we can have 3 independent sets. But here there are…

Q: How do I prove that A∖(B∩C)=(A∖B)∩(A∖C)?

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Q: Prove: If f₁, f₂ € R(a) on [a, b] with f₁ ≤ f₂, then so f₁ da ≤ f f₂ da.

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Q: Define f:R→R by f(x)=2x3+7.  Prove ff is a one-to-one correspondence. (i.e. a bijection)

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Q: Let X be a pre-Hilbert space then [(x, y)|

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Q: Cietni let ane let G4)= ti t2+34)j +4K HIC4) = 2,4し + 34j+(-144)1k fnd の) H4)XG(H) b) G4). HG4)

A: Given: Gt=ti+2+3tj+4kHt=2ti+3tj+-1+2tk

Q: Let f = u + iv be analytic in the connected open set and assume u and v satisfy u? + 3v2 = 4. Show f…

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Q: Evaluate f (x - y)(y - z) dV for the box B3 with dimension: [4, 9] × [0, 1] × [0, 1]. (Use symbolic…

A: We have to evaluate the integration .

Q: What is the domain of {(v²,y)|y€Z}? A Z B z? © V7 D Z except (0}

A: We have to find domain of {(y2, y) :y belongs to Z} i. e we are given that range set is set of…

Q: Prove: let v: X Y bijective then V inverse(v(x)) = X

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