Topic is Euclident n-space Let x,y,z ∈ Rn , and c ∈ R. Then prove the ff: a. ||x|| >= 0 b. ||x|| = 0 <=> x = 0 c. ||cx|| = ||c|| · ||x|| d. ||x · y|| <= ||x|| · ||y||

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A: The given set S=ℝ. The relation R defined in the set S is ∀x∀yxRy⇔y≥x+3. We have to check whether…

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Q: Let A = {x Rx ≥ 5 and x 8 and x < 12} (a) Find (AUB) nC. (b) Find An BNC. (c) Find Ce

A: Sets

Q: Let E and F are mutually exclusive and Pr(E) = 0.2 and Pr(F) = 0.6. Find Pr(E ∪ F).

A: Given :

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Q: Let G be the interval (-1/2, ∞). Let a be the operation on G such that, for all x, y = G, x x y =…

A: Introduction: In mathematics, an identity is a relationship between two mathematical expressions A…

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Q: Let v = (−1, 2, −3, 0, 5)T . Find numerical values of ||v||1, ||v||2, ||v||3, ||v||4, and ||v||5.

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Q: For r> 0 let A set A. = w:w= exp (¹) where 0 < |z| <r} ; determine the

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Q: Let x, y ∈ R such that x 0.001. Prove that |x · y| = |x| · |y|

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A: Given N :R^n→R be a norm on R^n.      To show : N is a convex.     Definition of norm :…

Q: (d) Show that T is bounded and find ||T||. (e) Find the adjoint T* of T.

A: We will use the vector/matrix notations to visualise things easily.

Q: If A⊂B, prove that if B is bounded, then A is also bounded. Is the converse true?(if not true give…

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Q: JJ Let Prove that V₁, V₂ and V₂ E IR || V₁ + V₂ +vall ≤ ||v₁|| + ||v₂|| + ||V3|1

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Q: ): Discuss the continuity of the function is : x² – 1 X 2 f(x) = 2х — 1 - x+1 at x = 2.

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Q: Let ƒ be continuous and nonnegative on [a, b]. Show that there is c € [a, b] with f(c) = (±a få…

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Q: Let {(x, y) E R² : |x| 1}. {(x, y) E R² : 0 < x < y}, {(x, y) E R² : 0 < y <¤ < 1}, U {(7, y) E R² :…

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Q: Prove the inequality |sina−sinb|≤|a−b|.

A: Consider the inequality

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Q: Let S and T be sets of real numbers, and define S · T to be {x. y: xe Sandy e T}. Show that if S and…

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Q: 2b. In R show that : || x - y || || x + y || ≤ 11x11 ² + 11y11²2² -(1

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Q: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = A O (A)° = A – int A %3D…

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

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Q: Show that both R and R2 may be covered by countably many open balls.

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Q: Let x = [1, -3, 2]T. Find ||x||₁, ||x|4, and ||×||∞·

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Q: A⊂B, prove that if B is bounded, then A is also bounded. Is the converse true?(

A: Let A⊂B, where B is bounded. Then there exist two real numbers k and K such that k ≤ b ≤ K for all b…

Q: Let G be the interval (-3/2, ∞). Let □ be the operation on G such that, for all x, y € G, xay=2xy…

A: Introduction: In mathematics, an identity is a relationship between two mathematical expressions A…