a) In the set of real numbers A = {x | | x - 2 | > 1, x E IR} cluster is given. Accordingly, express the set A as an interval and graphically represent it on the real number axis.

Q: Suppose that E and F are points on the number line. If EF = 20 and Elies at 4, where could F be…

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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(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: 1.-Determine the set of all points z such that they satisfy the following equation:

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Q: Use the Heine-Borel Property from (5.1), which states that a set of real numbers is compact if and…

A: See solution below. Here we used definition of bounded and closed sets.

Q: Let S={(a,b) : a, b \in Q} be the collection of open intervals in R with rational endpoints. Find…

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Q: Suppose that Y and Z are points on the number line If YZ = 16 and Y lies at 4, where could Z be…

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Q: Let n be a real number and let e be a positive real umber. Determine whether there exists a 1 real…

A: Given n be a real number and ε be a positive real number.

Q: 13) Use the Intermediate Value Theorem to show that the polynomi: zero in the given interval. g(x) =…

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Q: Let x be a rational number. Prove that if ry is irrational, then y is irrational.

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Q: Show that the Archimedean Property is a consequence of the assertion that every interval (a,b)…

A: Archimedian property : For every positive real number c there exist a natural number n such that…

Q: In the diagram, eleven circles of four different sizes are drawn. Each circle labelled W has radius…

A: Given problem is : In the diagram, eleven circles of four differentsizes are drawn. Each circle…

Q: 10. (a) Prove: If x and y are real numbers with x < y, then there are infinitely many rational…

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Q: Suppose that E and F are points on the number line . If EF = 11 and E lies at 3, where could F be…

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Q: Describe the set S = { x : I 1/2x - 3 I &gt; 4} in terms of intervals.

A: S=x : 12x-3&gt;4

Q: Given x > 0 and a real number y, there is an n in N such that nx > y.

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Q: Write down the formal definition of the real numbers "T and e. TT

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Let R be the set of all real numbers with the euclidean topology. Show that R has an uncountable…

A: Let R be the set of all real numbers with the euclidean topology. Show that Rhas an uncountable…

Q: How many different size of infinity are there? How does it relate to one-to-one correspondence?

A: There are actually many different sizes or levels of infinity; some infinite sets are vastly larger…

Q: Prove that interval [0,1] is uncountable.

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Q: Prove this: There is a one-to-one correspondence between any line and the set of real numbers.

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Q: In the set R of real numbers, define C, = (n,n+1) = {xeR|n < x <n+1} for each integer n. Determine…

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Q: If x is even and y is odd, show that (x + y)? is odd.

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Q: Prove that the boundary points of an open interval (a, b) CR are the points a and b.

A: A point a∈ℝ is said to be boundary point of a set A⊂ℝ, if Ba,r∩A≠∅ and Ba,r∩Ac≠∅ for every…

Q: If x and y are two positive real numbers, then there exists a positive integer n such that ny>x

A: Proof: Suppose there does not exist any positive integer n such that ny&gt;x then ny≤x   ∀n∈ℕ

Q: Prove that if x is a real number, then [2x] = [x] +

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Q: A set L cR is convex if:

A: In this question, we have to find the correct option for the convex set L⊂Rn  here we use the direct…

Q: A set consisting of single point has measure zero .A O nonzero measure .B O open .C O none of…

A: We have to find the measure

Q: x and y be two real numbers such that x + y is rational.

A: Given statement:   x and y be two real numbers such that x+y is rational. To Prove:  If x is…

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 a < b. Find the crossing point n0 at which c1na = c2nb, for any c1, c2.

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Q: Show that the interval [0,1] is a closed set in R

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Q: (a) By drawing a suitable diagram, prove the triangle inequality |z1 – 2| > |21| – |2|-

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Q: Let A; = [0,2+. Find U A; and N A; Answer in interval notation.

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Q: let a∈R. give an example of nested intervals of each of positive length such that the intersection…

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Q: Suppose that J and K are points on the number line. If JK = 18 and lies at - 3 , where could K be…

A: NOTE: Refresh your page if you can't see any equations. . the distance between J and K is 18 and the…

Q: Express the set of real numbers between but not including 2 and 7 as follows. (a) In set-builder…

A: Set: In set builder notation the real number between 2 and 7 but not including both the numbers, Let…

Q: Find the concavity intervals of Cari selang-selang kecekungan bagi f(x) = 2x³ – 9x² + 12x – 4

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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: For any integer m and any real number x, if x is not an integer, then [x]+[m-x] = m-1.

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Q: Let x be an irrational number. Prove that there exists an irrational number y such that xy is a…

A: Let x be an irrational number. The main objective is to prove that there exists an irrational number…

Q: Let R be the set of real numbers and let B be the collection of all intervals of the form [a, b),…

A: Consider the provided question, Given that, R be the set of real numbers and B be the collection of…

Q: Show that the class of closed interval [a,b), where a is rational and b is irrationals and a <b, is…

A: Base for topology: A collection B of subsets of topological space X is said to be base for topology…

Q: Is this set finite or infinite? {x € R|1 < x < 2}

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Q: Describe the set S = { x : I } x - 3 I &gt; 4} in terms of intervals.

A: S=x : x-3&gt;4 Now consider, x-3&gt;4⇒x-3&lt;-4  or  x-3&gt;4   ∵x&gt;a⇒x&lt;-a  or  x&gt;a⇒x&lt;-1…

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 real…

A: Given that A,B,C are sets of real numbers with C is dense in B and B is dense in A. We know that a…

Q: 1) Consider the set X = {x E R' / 93x1+34x2 + 26x3 + 18.x4 < 131, x1, x2, 3, X4 E {C (a) Show that…

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Q: m above, prove that for every real number c>0, there is a 1/N < c.

A: In every real number c, there is integer N which is N &gt; c.

Q: Suppose A is a set of real numbers such that 7EA and sup A >7. Must A contain a number grater than 7…

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Q: The Cartesian product of finite and nonempty sets can be illustrated by the intersections of a…

A: Given:  a=nA and b=nB To find: Explain why the number of intersection points in the crossing line…