prove using soundness that the converse of Axiom 2: (vx)A → A[x := t] is not a theorem

Q: Show that the function u(z, t) : satisfies ua = %3D %3D a2t2-2

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Q: 1- iz If z = x +iy and w =- z-i show that w = 1= zis purely real

A: Use modulus property

Q: Let z = ƒ(u, y), u = ax + by, and y = ax - by. Express zx andzy in terms of fu , fy , and the…

A: Given, Let z=f(u,Y) u=ax+byY=ax-by

Q: Prove Theorem (A + B )T = AT + B T

A: Prove Theorem (A + B )T = AT + B T

Q: Prove that Cov (x, y) = E (xy) – E (x) E (y).

A: Cov(x,y) is covariance of x and y. E(x) is expected value of x E(y) is expected value of y.

Q: Let x, y ∈ R. Suppose x^2 +y^2 = 0. Prove that x = 0 and y = 0.

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Q: X= R² and T = { $₁,x} V {₂\x>0} where U₁ = {(x, y) | x² + y ² < 2² } Prove (XT) is Topology

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Q: Let A={x e R|-1sx s1} = B, Show that f:A →B given by f (x)=x|x| is a bijecti on.

A: we have to show that given function is bijective

Q: If la,} and {b„} are divergent then {a, + b,} is divergent. True False

A: Given: an and bn are divergent . The objective is to state whether an+bn is divergent is true or…

Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann eguations if u, = v, and u,…

A: Thje given functions are,u(x,y)=y(x2+y2) , v(x,y)=x(x2+y2)

Q: Use Cramer's rule to solve for y without solving for the un- knowns x, z, and w. 4x + y+ z + w= 6 .…

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Q: Prove that r(z) = (In-)²-'dt , Re(z) > 0.

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Q: Use Theorem 2.4.1 to prove that f(x,y) = 1 - x^2 - y^2 defined in R^2 is concave.

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Q: Let z = ƒ(u, y), u = ax + by, and y = ax - by. Express z x and zy in terms of fu , f y, and the…

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Q: Prove: If f1, f2 € R(a) on [a, b] with fi < f2, then ' fi da < s' fz da

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Q: Prove that f-(U B: -U(B.), %3D i=1 i=1 and n N (B.). 3=D1 i=1

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Q: Prove that if a e R, then a 0 = 0 = 0 · a. Suppose x, y, z E R. Show that xz = yz if and only if…

A: We will prove the given statements.

Q: 1-a*** a zi i. La 1-a

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Q: Prove that the multiplicative inverse law is false for S. [That is, don't just write down a…

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Q: of at (x, y, z) = (2, 3, 4). ay Let f(x, y, z) = (x + y)z. Evaluate

A: Given :fx,y,z = x+yz

Q: y Verify Euler's theorem for u = x' In

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Q: Exercise 3. Show that f : Z → N so that f (z) = z² + 1 for all z E Z is neither one-to-one %3D nor…

A: One-One Function: A function is said to be One-One if for every element in the domain there exists…

Q: b) g(x) = S,* In(2r + 1)dr

A:

Q: Vr+2+Vx-2 If a = prove that a -ax +1=0 Vx+2-Vx-2

A: To prove , we need to simplify the left hand side of the equation till it becomes 0. To do this…

Q: Prove that Ix I - I y I ≤ I x - y I.

A: We have to prove that  x-y≤x-y We know, x+y≤x+y Then,  x=x-y+y⇒x=x-y+y≤x-y+y⇒x-y≤x-y

Q: Prove that ||A||1 = max1<j<n{£", |aij|}. %3D

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Q: (ii) Verify Cauchy-Goursat theorem for f(z) = 5 sin 2z if C is the square with vertices at 1土も-1土。

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Q: Let f(x, y, z) = x²y°z – xyz², and compute: (a)

A: In this question we will use basic of vector  calculus i.e gradient of any function and curl of…

Q: Let x, y € R" be given. (a) If ||z||||y||=1 and z-y = 1, show that z = y. (b) If ||z|| ≤ 1 and ||y||…

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Q: (b) Prove that the multiplicative inverse law is false for S. [That is, don't just write down a…

A:

Q: Prove that this argumentation is valid Vx. 3y. A(x, y) - Vx.3y. B(x, y) 3x. Vy. ¬B(x, y) : 3x. Vy.…

A: Given: An argument…

Q: Use Cramer's rule to solve for y without solving for the un- knowns x, z, and w. 4x + y + z+ w= 6 3x…

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Q: X = R² and = = { $₁x} V { \\x>0} where U4 = {(x,y) | x² + y² <²} Prove (x, t) is Topology

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Q: Let AC C([-1,1]) be defined by A = {f€c'(I-1, 1), f(0) = 0, f'(x)| < 1, V x€ (-1,1)} Prove that A is…

A: The set A is defined as A=f∈C1-1,1, f0=0, f'x≤1, ∀x∈-1,1. To prove that A is relatively compact.…

Q: Let x, y ∈ R. Suppose x2 + y2 = 0. Prove that x = 0 and y = 0.

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Q: Use Cramer's rule to solve for y without solving for x, z, and w. 5 x + y + z + W = -7 5x + 3y - z +…

A: Here we have to use cramer rule for solving the system of linear equations

Q: Let F I2). Evaluate JJaw as z² + y? 0

A: In the given question, the concept of the Divergence Theorem is applied. Divergence Theorem Assume S…

Q: LZHP - DTNV. Complete the statement below. ZZ = _2_

A: Consider the following condition: LZHP~DTNV Hence, ∠L≅∠D∠Z≅∠T∠H≅∠N∠P≅∠V

Q: 1. If x, y, z E R, prove that (a) \x+y\ < \x| + \yl- (b) |x– y| < |x – z|+|z– y|, and (c) ||지| -…

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

Q: 3. Prove F (Vr)(A→ B) → (Vx)A → (Vx)B. ->

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Q: Let a, y, z be positive real numbers such that r +y+z xyz. Prove that 1 1 1 1 1 1 1+ x2 1+ y2 1+ 22…

A: Given- x + y + z = xyz This inequality can be proved by magic no. Concept-  First we will prove…

Q: Prove that (X+Y) (X+Z) =X+YZ

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Q: on vectors for t2 0.

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Q: Compute the value of f.dr If F= -yi+xj and c is a from (-1,1Y to CI,1) %3D SemICirele

A: Proceed as shown below

Q: Prove using soundness (Required): (∀x)A → (∀x)B ⊬ (∀x)(A → B).

A: The proof is shown below.

Q: a- if w = f(u, v), u = x + y, v = x-y Prove that = ( - )* aw aw %3D ду du.

A: