) Give definition to the convex set of the space Rn. Also give a definition for when the function f : K → R, where K ⊂ Rn, is convex.   c.) Are the functions f(t) = −t ln t,  t > 0, and g(t) = t4 −t2,t ∈ R, convex? Justify your argument.

Q: 4. Let M be a set with a real valued function D M x M satisfying the following: (1) D(a, a) = 0; (2)…

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Q: Show that d given by d(x, y) = |x-ylis a metric on R by using definition 4.2.1. You may want to use…

A: we knew  that given metric d(x,y) = |x-y| is always ≥0.  now d(x,y) = 0 ⇔ |x-y| = 0…

Q: A: For each of the following maps T: V→ W, find the spaces V and W and say if they are linear or not…

A: The given map T:V→W such that Tx, y=1, y. We have to check whether the map is linear or not.

Q: What is the Brouwer fixed point theorem and how is it connected to the notion of retracts?

A: To Explain :  Brouwer's Theorem and how it is connected to the notion of the retracts.

Q: Consider the set V = {(x, y) = R² : x+y=1}. on this set defined in the following way: (x2, y2) = (x₁…

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Q: 3. Let M be an arbitrary set and define D D(x,x) = 0 for all x EM; for xy, D(x, y) te [1, 2]. prove…

A: Definition of Metric Space :   A Metric space is a ordered pair  M,D satisfying the…

Q: (a) Determine whether or not it is convex or concave. A = {(x, y) = R²|yx4, and -2≤ x ≤ 2} B = {(x,…

A: Necessary Sufficient Nature f'x0=0 f'x0=f''x0=f'''x0=..=fn-1x0=0  fnx0>0 n even Convex f'x0=0…

Q: Let f: X → Y and g : Y → Z be functions. (a) Show that if g of is one-to-one (or "injective"), then…

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Q: Let D, E be non-empty sets, let ACD, let BCE, and let f: D E be a function. Prove that Anf¹(B) ≤…

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Q: Let X = {1, (b) xm} and Y = {y₁, Yn} be finite sets. Determine the number of one to one functions…

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Q: 3. Use the definition of convex sets to answer the following: (a) Show that if the sets S and T are…

A: As per guidelines, I can do only first three subparts of the question.

Q: 3. Use the definition of convex sets to answer the following: Show that if the sets S and T are…

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Q: Determine whether the relation R on the set Z is reflexive, symmetric, antisym- metric, and/or…

A: Given: The relation ℝ on the set ℤ and x,y∈ℝWe need to check whether the given ℝ on the set ℤ is…

Q: Function f is defined as f (z) = z-6z+14. What is the minimum value of f ()? OA.-3 O B. 3 OC. 5 OD.…

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Q: R= {(0,2). (1, 1). (1,3), (2, 1). (2, 2). (2,3 For relation R find the Reflexive closure; Symmetric…

A: Given:- Set A={0,1,2,3}   , relation R:A×A is defined as : R={…

Q: pls explain to me step by step. pls dont skip any steps. thanks

A: Proof:The metric defined on R is d (x, y) = |x – y|.In order to prove the real line is a metric…

Q: 4. Define a relation R on the x-y plane as follows: Given two points (X₁,Y₁) and (x2,Y2) in the x-y…

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Q: Let us define a relation S on R3: If p1 = (x1, y1, z1) and p2 = (x2, y2, z2) are two points in R3,…

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Q: a. Prove that for any sets A,B⊆X and for any function f:X⟶Y, f(A∩B)⊆f(A)∩f(B).       b. Prove that…

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Q: Determine (with argument) whether the following subsets of R are sub- spaces: 1. {(a, b, c) E R³ | a…

A: By Bartleby guidelines I can only answer the first three sub-questions. please repost the question…

Q: Exercise 4.4.5. Let A and B be sets. Prove that there is a bijective function f: A × B → BXA.

A: A and B are two sets.  We need to prove there is a bijective function            f : A × B → B × A.

Q: Describe the set {TER²: doo (r,(0, 0)) = 1}.

A: To Describe: the set {x∈ℝ2: d∞(x,(0,0))=1}.

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: For each of the sets find the set of boundary points and the set of interior points. Also state if…

A: We have, (a) [0,1]U{4} (b) x,y∈ℝ2:x2+y2=1 Here we have to find the  o find: Interior Boundary…

Q: Let A E R2x3, be R². Determine whether or not S = {x R³ | Ax=b, x₁ ≤ 0, x₂ ≤ 0} X2 sa convex set.

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Q: For functions:f(x, y) = x − yg(x, y) = x2 + x − yDetermine whether the following sets are closed and…

A: Final answer:.a) Not bounded.b) Not bounded.c) Not bounded.d) Not bounded.Explanation:Explanation:…

Q: For nonempty sets A, and B, let f: A →A and g: A → B be functions. Prove if the statements that are…

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