Prove that if A ⊆ B and both sets have minima then minB ≤ minA

Q: Find all values of x such that fx>0 and all x such that fx<0

A: f(x)=-1x39-3 To find x values  such that f(x)&gt;0 we find out x intercepts To find x intercepts ,…

Q: Suppose A and B are each bounded above. Define the set A+Bas follows A+B = {a +b: a E A, bE B} We…

A: We know that A number ,α is called G.L.B or infimum of a set of numbers A if and only if α is a…

Q: Suppose A and B are bounded and nonempty subsets of real numbers and 1 E R. In this case: sup(A U B)…

A: Let A,B not empty, bounded subsets of R .Prove thatsup(A∪B)=max{sup(A),sup(B)}. Let: ⇒x∈A∪B⇒x∈A or…

Q: (a) Prove that 3n2 +1 is even if and only if n is an odd integer. (b) Suppose A, B,C and D are sets.…

A: The given question has two parts. The solution is explained below.

Q: 37. Which of these three statements is/are not true? I: Every finite nonempty subset of R has a…

A: We shall answer all the parts and provide justification as to why a particular part is not true. In…

Q: Show that gcd(a, bc) = 1 if and only if gcd(a, b) = 1 and gcd(a, c) = 1.

A: Consider, gcd(a, b) =1 and gcd(a, c) = 1. If gcd(a, b) =1 then there exist integers m and n such…

Q: Let S = {1,2,3,4,6,8,9,12,18,24} with the relation "x divides y". Which one is false? 1. O Tidak ada…

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Q: Find an example of a nonempty set satisfying: 1. Closed, Dense in R 2. Open, Dense in R ~1. & 2.…

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Q: 9/State the Sufficient condition for the best apfraximation ant Prove t

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Q: Let U = (0, 1) U (2, 3) U (4, 5) be a union of three open [0, 10] \ U. Show that A is compact. 2.…

A: Que  Ans : given         U = (0,1)U (2,3) U (4,5)  be unian              of 3 open sets    A= [0,10]…

Q: Prove or disprove: If A ⊆ B ⊆ C and A and C are countably infinite, then B is countably infinite.

A: Let A ⊆ B ⊆ C The sets A is countably infinite. Since A ⊆ B and A is countably infinite, it implies…

Q: Q 3 Prove or disprove each of the following: For all sets A, B and C, (а) (А — В) — О%3 (А- С) — В

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Q: Let P(x) be a property about some object x of type X. If we want to disprove the claim that "P(x) is…

A: Given that : Let P(x) be a property about some object x of type X. And the claim is " P(x) is true…

Q: Prove that if m | k then 2" – 1| 2k – 1

A: Use the definition of Divisibility, "a divides b" means a and b are integers and there is an integer…

Q: Let A and B be nonempty sets of real numbers. If sup A exists then: a. sup (A U B) exists if sup(B)…

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Q: Determine a and b such thatp(x) = x2 +ax+ bsatisfies p(l) = 0 and p' (1) = 4.

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Q: Let a and b be integers, not both 0. Show that there are infinitely many pairs s,t of integers such…

A: We will first show that  there are infinitely many pairs s,t of  integers such that :as+bt=0 Now, we…

Q: State and provié the Second isom theom

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Q: Let a and b be positive integers such that ax + by = 7 for some integers x and y. Then %3D

A: We have to choose  current answer of given statement.

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Q: Show that gcd(a,n)=1 if these exist m and k such that ma+nk = 1.

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Q: (ii) Determine supremum and the infimum of the set A=(0,10)∩R∪[−2,8]A=(0,10)∩R∪[−2,8] if they exist

A: It is assumed that R denotes the set of real numbers.

Q: Find all integer x satisfying the condition x = gcd(5x + 2 3r) lustify vOur ancu

A: The complete solution is in given below

Q: (i) Show that a non-empty subset of real numbers A is nowhere-dense if and only if the interior Aº…

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Q: Determine the minimum integer n such that any set of n consecutive positive integers contains at…

A: Hai

Q: Let S = {0, 1}. List all the strings of length 4 over S that contain three or more 1's. (Enter your…

A: 1. The given set is S={0,1}. It is also given that the set S contains all the strings of set 4 and…

Q: If a set E has an infimum, then it has only one infimum. Proof: Suppose that E has two infima, call…

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Q: Set A is bounded below if and only if set -A is bounded above. True False

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Q: *3. Suppose x is a real number and > 0. Prove that (x - e, x + e) is a neighborhood of each of its…

A: Given that 'x' is a real number and ε&gt;0. To prove that x-ε, x+ε is a neighborhood of each of its…

Q: Suppose A and B are bounded and nonempty subsets of real numbers and 1 E R. In this case: If A CB…

A: If you understand the concept of inf and sup then these statements are obvious. Remember that inf of…

Q: An inventory consists of a list of 115 items, each marked “available” or “unavailable.”  There are…

A: Consider the given information.

Q: Let a1, a2, a3, . .., an be real numbers. Prove that at least one of them is greater than the…

A: We prove it by contradiction.

Q: let 19, b) = 1, them prove that there exists x20 Such that ijzo ax_by = 1

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Q: Suppose A and B are bounded and nonempty subsets of real numbers and 1 E R. In this case: sup(AB) =…

A: Suppose A and B are bounded and nonempty subsets of real numbers and λ∈R. We need to prove that…

Q: Let A and B be nonempty sets of real numbers. If sup A exists then: O a. sup (AU B) exists if sup(B)…

A: If s is an upper bound for A∪B iff it is an upper bound for A and for B, so s≥supA and s≥supB   The…

Q: if the If the conditions of the existence theorem are fulfilled and the conditions of the conditions…

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Q: Prove that a non empty set A is countable if and only if there exist a bijection from A to a…

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Q: . Let A, B be sets of real numbers. We will say that B is dense in A if for any s e A and for any…

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Q: Prove Bernouilli’s Inequality which states: If x ≥ −2 then (2+ x)n ≥ 1 + nx for all natural numbers…

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Q: 10. Prove: A and B countable > Ax B countable.

A: According to the given information, Let A and B be two countable sets then it is required to prove…

Q: 3. Let A and B be effectively enumerable sets of natural numbers. Show that the sets AnB and AUB are…

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Q: 2. Consider a four-input function that outputs a 1 whenever an even number of its inputs are 0. a)…

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Q: One of the theorems is called well-ordered theorm: .a For any set A ,we have |A|<|P(A)| .b If S is…

A: The well-ordering theorem, also known as Zermelo's theorem, states that every set can be…

Q: Please solve part (a).

A: Hi! We are answering only the first two problems. If you need the solution of rest questions, then…

Q: 6. If k divides |G| then 3H ≤ G such that |H|=k. *True *False

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Q: Let a and b be integers, not both zero. Then a and b are relativelyprimeif and only if a)1 = ax +…

A: Option D will be the right answer.

Q: Please solve this problem.

A: Let V be the vector space and

Q: prove that if a>4 then (a-2)^2>4

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