Let U= set of all natural numbers A = {r:x = 7y, y E N}, B = {r:x = 4k, k E N}, and C = {r:x = 3n + 1, n E N}. List the smallest six elements in each of the following sets: AU (BUC){| } (AU B)' { [ }

Q: Suppose that X,Y and Z are subsets of {1, 2, 3, . . . , 10} and |X| = |Y| = |Z| = 7. (i) Prove that…

A: Given that X,Y,Z are subsets of {1,2,3,...,10} |X|=|Y|=|Z|=7. X∪Y≤10 i)X∩Y≥4 X∪Y=X+Y-X∩YX∩Y=X+Y-X∪Y…

Q: 3.2 Prove that in any metric space (S, d) every closed ball S,xo is a closed set.

A: Let (S, d) be a metric space. Consider a closed ball Srx0 with both center x0∈S and radius r>0…

Q: Let {X;} be i.i.d.~ Exp(0) and a; > 0. Denote by Z = min{a,X1,..., amX,}. Suppose that 0. Prove that…

A:

Q: (6) Let (X,M.) be a measure space, and {Ej}cM Prove that : (I) If E,SE,S…. ,then j (II) If E,2E2...…

A: The given problem is related with measure theory. Given that X , M , μ is a measure space and Ej1∞ ⊆…

Q: Let a1, a2,...,a, be real numbers that are all contained in the interval [-7 /2, 7 /2]. Use the…

A: 1. The pigeonhole principle states that if n items are put into k boxes with n>k, then at least…

Q: Suppose that E1 and E2 are measurable sets. Prove that m(E1 U E2) + m(E n E2) = m(E1) + m(E2).

A: We have to prove that the given statement.

Q: Use the following rule in mathematical morphology to fill the holes inside the image A to be black.…

A:

Q: be a collection of sets in S'. Show that we can express U E, as the disjoint union U, Er of sets in…

A:

Q: Consider two sets, A = {1, 2, 3, ..., 100} and B = {3, 5, 7, 9, ... , 201}. Prove that |A| = |B| by…

A:

Q: How many 5-tuples (x, y, z, u, v) of natural numbers satisfy x + y+ z+u + v = 20? Let x= X-1,…

A:

Q: 7. Suppose that X and Y are disjoint sets and k E Z>o. Prove that the function k so:U(P:(X) × x…

A: Let us solve step by step.... X and Y are disjoint sets...

Q: 2.28 let {Ei} i Ej be a Measurable pairwise dis joint sets. prove that for any bounded set A…

A:

Q: 4) Consider the collection of intervals on the real line: B = { (a, b) | a 1}. Is this a basis for…

A:

Q: Prove that ', 12 and lº norms satisfy the properties (N1)-(N3). (N1) (Positivity) ||x|| > 0, and ||x…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Prove the following statement for a collection of natural numbers x1, x2, ... , xn and the set I =…

A:

Q: Let . be the operation on the cartersian product of real numbers R • R such that (a,x) * (b.y) =…

A: Given (a,x)*(b,y)=(ab,x+y)

Q: Prove that on average, permutations of length n have H, cycles, where Hn i=1

A:

Q: 2) L Let A and B be di screte sets of R". Def ine as p(t )= (1-t )a + tb for atA and b€B. tER,…

A: Given: A and B be discrete sets of Rn. Also, it is defined as: Pt=1-ta+tb for t∈R. And, a∈A , b∈B.…

Q: Let A & M and μ*(A) < oo. st B₁, B2,..., BEC, k <co with Bin B „²(AAŬB₂) <« j=1 any two sets E₁ and…

A:

Q: (a) Investigate whether d: RxRR delined as d(z, u) = (- u) is a metrie on the set ol all real…

A: We investigate given distance function is metric on R or not

Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. (2) If AC B,…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

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

A:

Q: Let D be a Euclidean domain with measure d. Prove that u is a unitin D if and only if d(u) = d(1).

A:

Q: Q2: Let the set under consideration be N for each n EN, define u, = {n, n+1, n+2, ...} and let 1 =…

A:

Q: 8. Sketch the sets X = {(x,y) E R :x +y s 1} and Y = {(x, y) ER:-1sys0} on R. On separate drawings,…

A:

Q: 4. Let A Sketch the collection of points Ar for i in the unit square.

A: Let A=1101. Sketch collection of points Ax¯ for x¯ in the unit square. Matrices maps a polygon to a…

Q: 7. Let X be a set with |X| = n € Z4. Prove that for any set Y, |Y| = n if and only if there exists a…

A:

Q: Prove that on average, permutations of length n have H, cycles, where Σ Hn i=1

A: We will take an arbitrary set containing n elements and observe the required permutations.

Q: In the Triangle Inequality of Euclidean n-space: Prove ||x - z || <= ||x - y|| + ||y - z ||…

A: The triangle inequality, in Euclidean geometry, theorem that the sum of any two sides of a triangle…

Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A:

Q: 4. Let r E R" be such that r = (E1,... , En). Show that the norms ...+ En and = V are equivalent so…

A:

Q: Let R be the set of all real numbers and let N be the set of all positive .integers a) If T=…

A:

Q: Let S be the set of integers. If a, b E S, define aRb if ab &gt;= 0. Is R anequivalence relation on…

A:

Q: Let A = {x ER|a° e Q} be a set. Is A countable? Justify the answer by showing the explicit formula…

A: A is set of all real numbers whose cubes are rational.  Now for any real number b, the cube can be…

Q: 3. Prove that the image of a measurable set E under the translation 4 is measurable and m(E+a)…

A:

Q: Identify all nonempty set/s from the following A = {x|x € N, 2x + 7 = 3} B = {x|x € Z, 2x + 7 = 4} C…

A:

Q: Let X be a finite set and f: X → X. Prove that f is an injection iff f is a surjection. (Hint: If…

A:

Q: Let {An : n E N} be the indexed collection of sets defined by 1 2, 2+ 2n An Prove that: nMeN An = Ø

A:

Q: .2 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

A:

Q: Find x, y, and z such that x³+y³+z³=k, for each k from 1 to 100.

A: By adding the cubes of positive integers we solve the given problem.

Q: Consider a finite set S of points in the two-dimensional plane. A point p of S is called a covidiot,…

A:

Q: 6. If A has m members and B has n members, prove that A x B has mn members 7. Sketch each of the…

A:

Q: Let k e Zt. Let A be a set such that |A| = k + 1, and let a e A (i.e., a is a fixed but unspecified…

A: Since you have posted a question with multiple subparts, we will solve the first three subparts…

Q: Let n be a positive integer and let R = {a1, a2, a3, ... , an} be a nonempty set where a; are real…

A: GIVEN: R = {a1, a2, a3,........, an}

Q: Let fELoi Show that i for any E>o, the set 3- S has a positive measure -

A: For f∈L∞ , f∞=infM&gt;0:μx:fx&gt;M=0For M&gt;0, defining the set AfM=X:1 fx&gt;M⇒For f∈L∞,…

Q: (b) Let (X, >, ) be a measure space. For A, Be ). Show that μ(AU B ) - μ(Α) + μ(Β) -μ(An ). %3D

A: Since you have asked multiple question, we will solve any one question for you. If you want any…

Q: 2. Let T={USR: U = Ø or U= R or U=(-∞o,a) for some a E I}. Prove that T is a topology on R (set of…

A:

Q: Prove that S UT = SN T for all sets S and T using the rules for complements of sets.

A: rules of complements x∈A⇔x∉A¯

Q: V7 In Exercises 3 to 8, list the four smallest elements of each set. 3. (2rx E positive integers} 4.…

A: Given :- {y|y=2x+1 ,  x∈ natural number} To find:- List four smallest elements of sets