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

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: 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: Which of the following sets is a function? * {(a, a)b,b)\a,c)} {(a,b){b,cXc,a} Option 2 О Оption 4…

A: We have to identify which of the following set is a function.

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

A: If a set has no element then the set is called null set. Null set is also called empty set. It is…

Q: |x²| = p^ n

A: Given: X=pWe need to show that Xn=pn

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

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Q: How many functions map {1, 2, 3, 4, 5, 6, 7, 8} onto {1, 2, 3, 4, 5} (i.e., how many surjections are…

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Q: Prove that la + b + c| < ]a] + ]b| + |c[

A: Prove that,    a+b+c≤a+b+c

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

A: Given that,f(z)=z+1zz=1

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: 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: 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: 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: . 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: _The Cauchy-Schwarz Inequality states .+ a%)(b} + b3 + ...+ b,) (a,b1 + azb2 + ...+ a,bn)? < (aỉ +…

A: Given

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: mutually disjoint? Why or why not?

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

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: The universal set is U with n(U)= 70, A and B are subsets of U with n(A)= 55, n(B)= 20 and n(A’…

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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: 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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Q: Let N={1,2,3,4,...}. Let P(x,y) be the predicate that x is less than or equal to y^2

A: (d) S = {1, 2, 3, 4, 5, 6, 7, 8, 9} (e) T = {3,4,5,6,......} (f) R = {} (empty set)Explanation:(d)…

Q: Let A and B be finite sets satisfying |A| = 6, |B| = 7 and |AN B| = 2. %3D Calculate the following:…

A: # we are entitled to solve three subparts at a time, please resubmit the other parts if you wish to…

Q: List the elements for the given sets {x|x is a perfect square FROM 1 to 10} {x|x is an integer…

A: Set: A well-defined collection of different objects. For example: Set of natural numbers= {1, 2, 3,…

Q: How many injective functions have domain [10], codomain [10] and are not surjective?

A: Ans : 0 (zero)

Q: Show that the set A = {n^2/ (n+1): ne N} is unbounded in R.

A: To prove that the set A={n2n+1:n∈N} is unbounded in R,

Q: Prove. For any x, y ∈ R and x < y, ⟨(x, y), ≤⟩ is not well-ordered, where (x, y) is an open interval…

A: To Prove: x,y,≤ is not well ordered, where x,y is an open set with the usual order "≤".

Q: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} onto {1, 2, 3} (i

A: According to the question, To find the number of surjective (onto) functions from a set of size 10…

Q: Which of the following sets is not closed for the operation °, which is defined by the rule a°b = a…

A: Option (a) is not closed under the operation °.Option (b), (c), and (d) are closed under the…

Q: Choose the correct option. The number of onto function from the set {1,2,….,11} to the set…

A: Let the set A be {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} and the set B be {1, 2, 3, 4, 5, 6, 7, 8, 9,…