Let x ∈ R ^n and r > 0. Is the set B(x, r) = {y ∈ R^ n : ||y − x|| ≤ r} a closed set in R ^n? Y/N

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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: 7. i) How many functions are there from the set {1, 2, 3, ...,k} where k is a positive integer, to…

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Q: If then the domain D of the function f is given by the set Select one: ○ {(x,y)|y² > x, ○ {(x,y) y²…

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Q: Let A = {4n^2 + 1|n ∈ Z}. Show that A is a subset of Z but not equal.

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Q: (c) Let X, Y and Z be sets. Prove that (X\Y)n (X \ Z) = X \ (YUZ).

A: We prove the given statement by using set theory.

Q: C) If an expression is given that A+A'B'C=A+B'C, find the minimal expression of the function F =…

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Q: X, Y and Z be subsets of the set . Verify the following propert XU(YUZ) = (XUY)UZ, XN(YnZ) = (XnY)…

A: X, Y, Z be subset of a universal set Ω to verify…

Q: prove for all x,y in R^n: (1+||x+y||) ≥(1+||x||)/(1+||y||)

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Q: . Is it true that the null set is a subset of every other set? (Y/N)

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Q: EX: Show that the telation ? >' is PartiaL oldering set of Por R-

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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: (a) Let M = R. Give the radius r and the center c of B(-2,5) n B(6, 7). ne

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Y Show that f (2)= 2+ Con formal maps %3D on %3D

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Q: 4) Let a < b be real numbers. Prove that the half open/half closed interval (a, b) is both an Fo and…

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Q: XA and x & f(x)}. Show Cim ƒ.) 38. Let a, b, c, and d be any real numbers such that a < b and c < d.…

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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: 2. Let N = {1,2,3,...}. Are the following functions injective, surjective and/or bijec- tive? (i) f:…

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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: d) Suppose X = {1,2, 3} and Y = {5,6}. The number of elements of the set P(X× Y) =

A: Elements in X = 3. Elements in Y = 2 Elements in X × Y = 3 × 2 = 6

Q: what is a proper space

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Q: . Sketch the graph of the set defined by A x B where A is the closed interval [-2, 4] and B = {1, 3,…

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Q: Let A, B C X be subsets and f : X → Y a function. Show that ƒ(AÑ B) ≤ f(A)Ñ ƒ(B)

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Q: O P(0) = {0} © P(A°)\ {0} C P(A)°

A: Hey, since there are multiple questions posted, we will answer the first question. If you want any…

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

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Q: V = { (x1, x2, x3) E R^3 |x1 2 0, x2 + x3 = 5 } of the set R^3 a sub of Show whether it is a space.

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Q: 3) Let X be a set with > 3 elements. Show that there exist bijective functions f :X → X and g : X +…

A: Let x=1,2,3 then, f=123132g=123321 `

Q: What is the norm of a partition of a closed interval?

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Q: PROVE THE STATEMENT "There exists a, b ∈ Z+, such that if a > 1 and b > 1, then gcd(a, b) > gcd(a^3,…

A: Given statement is wrong because there is no a, b ∈ Z+, such that if a &gt; 1 and b &gt; 1, then…

Q: Write the following sets in the roaster form. (a) A = {x | x is a positiveinteger less than 10 and…

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Q: Justify rigorously the given set is open in R³. (a) A = {(x, y, z) |x>0, y 9}. (b) B = {(x, y, z) |…

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Q: Prove that a closed convex set in R" has a finite number of extreme points if and only it is the…

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Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

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