Suppose u, v belong to V. Prove that = 0 if and only if ||u|| ≤ ||u+av|| for all a belong to F

Q: Prove that P(XY) = P(X)n P(Y) for all sets X and Y.

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Q: 1. If f is a function on a set A = {1,2,3,4,5} such that f = {(1,2), (2,3), (3,4), (4, x), (5,5)}.…

A: Answer C

Q: +12x+19 in the interval [-1,6] of the above no zeroes in the interval [+1,6] 5 zeroes in the…

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Q: What is the global minimum of the function, x² 5 f(x) 3 + 2 on the interval [-2, 2] ? a) 2 b) 23 3…

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Q: If A+ iB is Hermitian (A and Bare real) show that [ AB -BA] is symmetric.

A: As per the question we are given two real matrices A, B such that (A + iB) is Hermitian. Now we have…

Q: (b) Let {Ai e I}. Prove that 4) =UA;. |A, iel

A: We need to prove that ∩i∈IAic=∪i∈IAic  for set Ai/i∈I We will prove using the following method: To…

Q: Prove the Cauchy-Schwarz Inequality, ∣u ∙ v∣ ≤ ∣ u∣∣  ∣ v ∣∣.

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Q: 2. Answer the questions with Yes or No, and then explain why yes or why not. Justify your answer. a)…

A: The solution of given problem is given as follows :

Q: 1. Let A be an uncountable set and B C A such that there exists a surjective function f : Z+ → B.…

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Q: Using contradiction method, prove that for every nonempty sets A and B, An (B- A)=0.

A: The given problem is to prove the following set identity using the contradiction method.

Q: Cg + చ = చ

A: In measure theory, there is a general result known as the additivity of integrals that states that…

Q: b) A function g:Z→Z is defined by the rule g(m, n) = m + n Determine whether it is onto or not...

A: A function f:A→B is said to be onto or surjective if for all element b∈B there is an element a∈A…

Q: Suppose both f(z) and the conjugate of f(z) are holomorphic on a connected open set.  Show that f is…

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Q: Assumed that F OG = Ø, show that we can express the infimum and supremum of F NG in terms of the…

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Q: b) A function g: Z → Zis defined by the rule g(m, n) = m + n Determine whether it is onto or not..

A: We will answer the first question as we don't answer multiple questions at a time. Please resubmit…

Q: Prove that B = (0,1) is NOT a compact set.

A: To prove that (0,1) is not a compact set.

Q: Show that R is not compact.

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Q: 20 - If f is a function on a set A= {1,2,3,4,5} such that f= { (1,2), (2,3), (3,4), (4,x), (5,5)}.…

A: Use the property that in bijective functions each element of doamin and codomain has image and pre…

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: Find two examples for equivalet norms on three dimensional real space R³. .

A: To take two vectors whose norms are the same.

Q: Prove or disprove the following statement: A) Every T'3-space is T4-space?( explain your answer)

A: The statement "Every T3-space is a T4-space" is true. To understand why, let's first define the…

Q: Show that the function f: [1, 2] → [1,8], f(x) = 7x - 6 is a bijection. What relationship exists…

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Q: A = { 3, 0, 1 } B = { y, x }  what is the cardinality of A x P(B) ?

A: Given that, A = {3,0,1}  B ={ y , x }  Power set of B = {{},{x},{y},{x,y}}  Generally cardinality is…

Q: Let A = {1, 3, 5, 7, 9, 11}, B= {a, b, c, d, e}, and ƒ be the function from A to B defined as below:…

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Q: Let N R R be any norm on Rn. Show that N is a convex function (give all the details for the proof).

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

Q: 2) Let R real numbers, define = x,Y1 +4x2y2 + 6x3Y3 for x = (x1,x2, X3) and y (y1,y2, Ya). Is inner…

A: Let V be a vector space over ℝ.An inner product <·,·> is a function V×V→ℝ with the following…

Q: [Falkner Section 11 Exercise 15(b) – modified] Let S and T be sets. Prove that S and T are disjoint…

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Q: Prove that {2, 3, 4, ...} is countably infinite. You must prove that your function is bijective

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Q: Show that if gcd (a, y) = 1 and a \ xy Then a \ x

A: if 1. gcd(a,b)=1then there exist some integers m and n such that am+bn=12. suppose a and b are…

Q: Calculate ∪∞n=1 An and ∩∞n=1 Anwhere for each n ∈N the set An is defined as:…

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Q: Prove that homotopy with respect to a domain 2 is a transitive property.

A: We will show that homotopy w.r.to the given domain is a transitive property.

Q: For a set A, let {0,1} A be the set of all functions from A to the two point set {0,1}. Prove that…

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Q: Show that if a Banach space X is non-trivial, X + {0}, 3). then X' # {0}.

A: According to the given information, it is required to show that:

Q: Use only the definition of affine dependence to show that an indexed set {v1, v} in R" is affinely…

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Q: ) Let S be the non-empty set containing all a for which f(x): = monotonic for all xe R. Find S. 4a-7…

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Q: Q1) Prove or Disprove: Let L(F) be a linear space and A,B are two convex subsets of L. Then, AXB is…

A: The problem is related with convex set. Given that,                 LF  is a linear space  and A, B…