(2) Let X=R. Then T= {uSR:if xEu, then -XEUS is topolosy on R. (a) Write dewn open set in (R) which is an interval and containing the number -6 (othir than R). (b) is [o,5] closed in (RZ) ?(Justifd your antwer) (C) Find Int (z) where 7 is the set of integers. (Justift Your onswer)

Q: Can you prove that it is injective by contrapositive? Also, how is f surjective when y exists in R?

A: Different equivalent definitions can be given for injectivity. The function f:A→B is injective if…

Q: True or False? 11. If Γ = {[0, 1/n) | n ∈ N} then ∩Γ = [0, 1). 12. Let R be a relation on a set A.…

A: 11) if Γ ={[0,1n) | n∈N}  then ∩Γ=[0,1)         Γ= {[0,1), [0,1/2), [0,1/3), [0,1/4), [0,1/5),…

Q: it X- ( a,b,c,d) A= (a,c) B= (b,d), T= (X,0,(a ,b) .lc,d}} find Int (A) Int (B)

A: Interior of a set A is defined as the union of all the open sets contained in A. so in this we see…

Q: 2. Prove the general formulas (GF)» = G«F* and (F-')» = (F«)- in the special case where F and G are…

A: Given G and F re isometries in ℝ3 To prove  1.GF*=G*F*2. F-1*=F*-1 Since F and G are isometries in…

Q: 1.85 True or False: Finitely many of the open sets in the collection { #) |- wa} * 3r 2' 2 would…

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Q: Which 4-tuples a 0<a<b<c<d<6}. are int the relation {(a, b, c, d)|a, b, c, d are integers with

A: We need to identify the tuples of the integer relation .Let .

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

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Q: To show this, let f, g be polynomials in R[x] such that g-f = (x-3)h for some polynomial h in R[x].…

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Q: on Give an crample of a dis continous bijection fo[at] = [1,2]

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Q: Shafe froe fabe, oting your ener. or The sef of natural nurnkers Bith the cwuel metir is compact of…

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Q: Find the first 4 terms in the maclaunin serie, for Y=x/sqart (1-x^2)

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Q: x* - 2x3 + 2x² – 2x + 1 х — 1

A: This question is related to division of polynomials.

Q: If X is inner product space over real numbers then the inner product always greater than or equal…

A: The objective is to determine whether the given statement is true or false. Given statement is, " If…

Q: 5. Let F= [3,2,1] and r= [2,1,–3]. Compute the following: %3D a) Proj „F. b) Orth rF = F – Proj „F.…

A: We know that  ProjrF=r.F∥r∥2r Now, r.F=23+12-31=5 ∥r∥2=r,r        =2,1,-3,2,1,-3        =22+11-3-3…

Q: What is true about the function *) - as *-* ? A) B)(x)→0 from below OC) f(x) is undefined OD)x)→…

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Q: Let filN >N be defined by f(n) = Ź v(d) for all nEl din where r(R) is the number of the positive…

A: A multiplicative function on the set of natural numbers is a function T such that Tnm=T(n)T(m)…

Q: Let T be a topological construct on R defined by the form T= {Q₂R} U² E₂ = (a, ∞o) a € R} Find…

A: Let T be a topological construct onℝ defined by the formT={ϕ,ℝ}∪Ea=(a,∞);a∈ℝ

Q: For the following sets, determine its cardinality.   (ℝ^2)\(ℚ×ℤ)（ is Cartesian product）

A: We find cardinality.     (ℝ^2)\(ℚ×ℤ)（ is Cartesian product）

Q: Theorem 1 (For open convex domain). Suppose 2 is a convex open set. Let f be analytic in 2 \{a} and…

A: We can break the integrand {\displaystyle f}, as well as the differential {\displaystyle dz}…

Q: #4. Set XyZ F(x4a)= | 4 3ゃex xy tyz+zX メ*ytを Show that F is 11 an open set ontaining (1,0,-1). in O…

A: Given - Let Fx, y, z = xyzxy + yz + zxx + y + z To Show -  Show that F is 1 : 1 in an open set…

Q: y = C¡e e-2= + Cze³* + C2e%

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Q: f is Riemann integrable iff f is bounded and f has finite set of discontinuous points.

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Q: Show the the Sar induce

A: According to the given information, it is required to show that equivalent norm on a vector space X…

Q: graph the inequality on a coodinate plane x &gt; with line under it 2

A: To graph the given inequality x≥2.The equation x=2 has a graph that is a vertical line with all…

Q: Qend Jamain of functon

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Q: At what wräluer of * id fG)= メ-メ-2 ズ- 4 discontinuous?

A: The function will be discontinuous if the denominator is 0.…

Q: 학과 학번 Determine the value(s) of d satisfying that Lacos(br+c) + d de lim 2-0c tan r dr for any real…

A: Integrate as follows.

Q: prove that integer V intgers is an

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Q: The relation R=D(x,y): 5x + 4y >= 100)on the set of positive integers is : O a. Reflexive O b.…

A: R={(x,y):5x+4y≥100}

Q: (−1)” - {1- (-²; n = N / is n The supremum of the set S=1- a) 1 b) 0 c) n d) 2

A: We haveWe have to find the supremum of the set S.

Q: 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: Is N closed over subtraction? (Y/N) Is Q closed over multiplication? (Y/N) Is N closed over…

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Q: X-Geometridp=2) Let a. P(X> 3) b. P(X>3|X ≤4)

A: GeometricWe have to find given the probability.

Q: Which of the following sets are countable? ORIN {7,99, 512}. 1 1 1 .}. " 10 100 1000 The set of all…

A: We know R is uncountable and N is coutable.  So, R \ N is uncountable [as R = (R \ N) U N, if R \N…

Q: A Moving to the next question prevents changes to this answer. Question 2 Let AcR be bounded and set…

A: Choose the correct

Q: Ax) = ,5-9,4+73 -5,2 -8x+1 is continuous on (-o, co) because fx) is a polynomia and polynomials are…

A: The objective is to state true or false of the above question.

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.