State whether the given set E is bounded or unbounded and justify your answer with a proof. E = {z € Z : z = 7 mod 9}

Q: Give an enumeration procedure for the set L = {ab*a™: n> 1, m >1}.

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Q: In each of the following, draw a Venn diagram for sets A, B, and C that satisfy the given…

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Q: Let x E Z and y E Z. Prove the following statement: If y = 2 mod 4, then xy 1 mod 4.

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Q: Decide with a proof that the set Nk is countable or not, where N is the set of natural numbers and k…

A: According to the given information it is required to proof that the set

Q: Recall that P(n, r) is the number of r-permutations of an n-element set. Prove the following…

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Q: 2. Classify each set as closed, open, or neither. Justify your answer. A= {-+:nEN} E. 1-:] [0, 1)

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Q: Let S={xEZ:x+2=6(mod 15)}. Then Sis is the empty set a countable set is an uncountable set

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Q: Let s={xEZ:x+3=4(mod 20)}. Then S is* a countable set is an uncountable set is the empty set

A: Since Z is countable. Every subset of Z is countable. Thus S = { x in Z , x+3=4 (mod 20)} is also…

Q: Let A be a set with |A| = n. Using induction, prove that |P(A)| = 2^n .

A: To show that the cardinality of power set of A is 2n when given that cardinality of A is n.

Q: State the formula for the Principle of Inclusion and Exclusion for 3 sets. Draw a Venn diagram…

A: The formula for The principle of inclusion and exclusion  for three sets A |A\cup B \cup C| = |A| +…

Q: R= {(a,b)|a E T,b e T, a has higher GPA than b}. Why?

A: No, it is not a poset.  Since we know that a relation is a poset if the relation is…

Q: Compute, without proofs, the suprema and infima (if theyexist) of the following sets:(a) {m/n : m, n…

A: We have given the set {m/n : m, n ∈ N with m < n}. For which we have to find the infima and…

Q: Determine the supremum of the following set with full justification, if it exists. п — Sn = {an | n…

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Q: Consider the set T = {..., −34, −17, 0, 17, 34, ... } with the usual definition of less than (a0).…

A: Completeness property of real numbers:  Every non-empty set of real numbers that has an upper bound…

Q: Let A = {r € Z | r mod 15 = 5} and B = {x € Z| r mod 3 = 2}. Give an outline of a proof that A C B,…

A: Suppose we have two non-empty sets A and B . If x∈A then x∈B for all x∈A then A⊆B . If x∈B then x∈A…

Q: Let (S „}nEN is a seqeuence of measurable sets ,then: n)nEN 00 00 씨 US µ (S ) =D1 n=1

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Q: How many proper subsets does the set T={h,f,k} have

A: A set is given by, T = h, f, k. The objective is to find the number of proper subsets the T have.

Q: Let *' be defined on the set N. Which of the following are both commutative and associative?

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Q: Use the tabular method to to prove the following in general: 1. (An B) C (AU B) 2. The O is a subset…

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Q: Suppose a set A is uncountably infinite, that U is countably infinite, and that UC A. Show that A -…

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Q: 9. Give a proof by contradiction to show if A and B are sets, then A N (BN A') = {}.

A: Suppose A and B are sets.

Q: Consider sets A = {n E Z:n is odd} and B = {n E Z : n = 7k for some k E Z}. Prove that A and B have…

A: Prove that A and B have the same cardinality.

Q: 1. Let A = {z € Z:z = 2 (mod 5)}, and B = {y E N : y² < 90}. Compute the sets ANB, and B\ A in…

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Q: If m, n, p, q are integers, n > 0, q > 0, and r = m/n = p/q, prove that

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Q: prove that i } X is a Coun table set and P: X>Y is asurjecbive then Y is Countable

A: Note: X denotes cardinality of X, that is number of elements in the set X. If X is countable means…

Q: 3. Let M and N be nonempty set such that M = N, then |M| = |N|. %3D

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Q: Show that the sete S = {x E R]x = m/3, where m E N with m > 4} is countable

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Q: 4. Let E CRd, and for each nE N, define On as the set On z d(x, E) 1/n) Recall that d(x, E) inf{| -…

A: To prove the openness of the given sequence of sets

Q: Recall that for a set S, P(S) is its power set. Given two sets S and T, prove that S C T if and only…

A: Let S and T be two sets and P(S), P(T) are its power sets.Power set of a set is defined as the…

Q: (3) Let S = {; Find with prove :m eN,n E N}. %3D suprenum of S. (4) What is an unbounded set?

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Q: 106. Let a e Z, and let M, be the smallest inductive set containing a. Then Ma = Zza.

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Q: let to be anonempty subset of abounded set E-Show that infESinf to Ssup E ssup E

A: Given :  E0 is a non empty subset of a bounded set E. To show :  inf E ≤ inf E0 ≤sup E0 ≤ sup E

Q: Given any finite set A, (of size n), compute and compare |P(A x A)| and |A4|.

A: Please check the next step for answer

Q: Let S be a 50-element subset of {1, 2, 3,...,87} and let T = {s+11 :sE S}. Prove that SnT+ Ø.

A: This is a question of Set theory, application of the Pigeon Hole Principle.

Q: Let S(a, b) = {na + mb : n, m ∈ Z}. For any a and b in Z, not both zero, the set S(a,b) is precisely…

A: Take the set S={ax+by&gt;0:x,y∈Z} Without loss of generality, assume that a&lt;b. Then b−a&gt;0 is…

Q: Prove that the set S = {n² : n E N} is denumerable.

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Q: 1. Suppose A, B, and C are sets, A. Prove that: AX (Bn C) = (A X B) n (A x C) %3D

A: As per our guidelines we are supposed to answer only one question.Since you have asked multiple…

Q: Let, aj = 3 and for n > 2, an = 2an-1 +1, express a, in terms of n.

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Refer to the Venn diagram below. How many elements are in the set AUB? nAUB) -

A: Given,

Q: Let A be uncountable and B be countable. Prove that A\B is uncountable.

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Q: The sets A and B have 6 and 9 elements respectively, such that A is proper subset B , then the total…

A: Given that set A and set B have 6 and 9 respectively. Now that A is a proper subset of B which means…

Q: Show that if A is a countable set, and B CA is infinite, then B is countable. [Hint: How could we…

A: To prove that  If A is a countable set and B is a subset or equals to A is infinite then B is…

Q: Define the sets D = In E N} n+1 Prove whether D is finite, countable, or uncountable?

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Q: question 7

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Q: Prove by contradiction that for any sets A and B , if A∩(B∪C)= ∅ and B⊆C, then B-(A∪C)=∅.

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Q: .Which of the following sets is countable and which is uncountable: A = {x |x = 2n, n e N} B = {x |…

A: We will find out the required values.

Q: Explain why the Cantor Set is the set of all points in base 3 that have a repre- sentation using…

A: Given: We have a cantor set in the set of all points in base 3. To prove:- We can Represent the…

Q: Consider the relation = on the natural numbers 9...24. How many sets are in the partition of 9...24…

A: As we that '=' is the equivalence relation then we have a partition with the set of the natural from…

Q: Let {$,}nEN is a disjoint seqeuence of measurable sets , then: 00 00 n=1

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