Suppose ACB and A Ø. Prove that (a) if B is bounded below, then so is A, and inf A ≥ inf B. (b) if B is bounded above, then so is A, and sup A ≤ sup B.

Q: Let A and B be bounded NON empty sets of R, and let A + B := { a+b: at A, bt B3. Prove that supla…

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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 that A and B are subsets of R which are nonempty and bounded above, and suppose that BCA.…

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Q: If 0 ≤ an ≤ b, and Σb, diverges, then Σan diverges.

A: Given: The series an and bn in the interval 0≤an≤bn and if the series ∑bn diverges, then ∑an…

Q: Let a, b c and d be real numbers. Show that ab-cd = b(a-c)+c(b-d), and deduce that |ab - cd| ≤…

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Q: Let x E A. Prove that A is an infinite set if and only if A≈ (A - {x}).

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Q: let A be a bounded. Sup A non. n-empty set in R show that is a limit point of A.

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Q: Suppose S and T are nonempty sets of real numbers such that for x in S and y in T, we have x < y.…

A: Upper bound: Let A be a non-empty set. Then, a real number s is said to be the upper boundof the set…

Q: a) Prove that P(EUF) ≤ P(E) + P(F).

A: Given that Prove that P(E U F) ≤ P(E) + P(F)

Q: 5. Let A = {x E Z|x = 12r - 5 for some integer r} and B = {y € Z|y=6s +1 for some integer s}. Prove…

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Q: 9. Given an infinite collection An, n=1,2,... of intervals of the real line, their intersection is…

A: Let An : = [ n ,∞)  where  n = 1,2, 3 .....  . So { An } n∈ℕ is an example of infinite collection…

Q: Va Abe B: a <b

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Q: S is a nonempty set of real numbers that is bounded.

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Q: Let S be a bounded subset of R. The real number a is the supremum of S iff it is an upper bound for…

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Q: If A is bounded below and B is bounded above, show that inf(A − B) = inf A − sup B

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Q: Which of the following statements is always true ? Select one: O a. If A, B, and C are sets with AC…

A: Recall countability and uncountability

Q: Ex 4.) Let Sbe a non empty bounded set in 1. aj Let azo, and let a S:= {as: SES}. Prove that inf…

A: Let S be any -empty bounded set in a) Let  , and let  Since,   is a bounded subset of , Sup and Inf…

Q: Suppose that A is a nonempty set of real numbers, and suppose that a ER is an upper bound for A.…

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Q: 3. Prove that if fog is injective, then g is injective.

A: Injective Function:          A function f : X →Y is said to be injective ( or one-to-one) if, for…

Q: If Do, D1,..., Dn... nEN, are dense open sets of reals, then the intersection D = 0 Dn is dense in…

A: Condition: Dn is the open dense sets of reals, for all n∈ℕ. To find: Intersection of infinite open…

Q: Determine if the given subset W of the appropriate R" is a subspace of Rn. Justify your conclusion.…

A: The given subsets,(a) .(b) (c) The objective is to determine the subset  is a subspace or not.

Q: What is the interior of S? b. What is the boundary of S? c. Is S open, closed, or neither? All of…

A: a. Interior point is empty b. Boundary of set S is all point of set S c.  Set S is closed.

Q: For nonempty sets A, B, and C, let ƒ : A —> B and g: B-> C. Prove: If g of is one-to-one, then f is…

A: A function is said to be one-to-one if and only if  implies  for (1) Prove by the direct method…

Q: The greatest common divisor c, of a and b, denoted as c = gcd(a,b), is the largest number that…

A: Here is the solution

Q: Let A, B be bounded subsets of R. Which of the following statements follow (tick all that apply)? If…

A: Since A⊆B, every element of A is less than or equal to every element of B. Therefore, supB must be…

Q: Let a₁,..., an EC with |aj| < 1 for all j = 1,...,n and A₁,..., An ≥ 0 real numbers satisfying 1 j =…

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Q: True or false (PROVE YOUR ANSWER) :-If A is a set of real numbers that is bounded above and sup A∉…

A: The statement is True.   Proof: Given that A⊆ℝ is bounded above and sup A∉A Let b=sup A Then for any…

Q: Let a, b € Z. Prove that if a ≥ 11, then af b or ał (b+7)

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Q: Given that A is a subset of B, prove that the sup A is greater than/equal to sup B and inf A is…

A: Let A is a subset of B.To Prove:The sup(B) is greater than or equal to sup(A) and inf(A) is greater…

Q: Let S := a + √2 b + √2 : a, b EN Prove that S is countably infinite.

A: To prove that the set S defined as S={a+2b+2|a,b∈N} is countably infinite, need to establish a…

Q: Prove that if A = {x|x2 2 or x > 0} then (A, B) satisfies properties 4 and 5 Property 4: if b∈B and…

A: We are given that       A = {x | x2 &lt; 2 or x &lt; 0}       B = {x | x2 &gt; 2 or x &gt; 0}   Now,…

Q: Let A, B be bounded subsets of R. Let A+B = {r+y:1 € A, y E B} and AB = {ry : 1 € A, y € B}. Which…

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Q: B. Let S and T be sets of real numbers, and define S. T to be {x.y:x e Sandy € T}. Show that if S…

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Q: For any X⊆R and z∈R define X+z to be the set of all x+z where x∈X. Prove that if ?s is the supremum…

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Q: If X is uniform on [2, 6]. Find a. P(X > 3), b. P(4 ≤ x ≤ 5)

A: X~U(a,b) probability density function pdf of uniform distribution is.  f(x)=1/(b-a)  , a&lt;x&lt;b

Q: Let i, r, and k be positive integers, and let s gcd(i, r). Assume that r | ik. Prove that | k.

A: To prove the divisibility property under the given conditions

Q: Let a and b belong to Q with b ≠ 0. Show that Q(√a) = Q(√b) ifand only if there exists some c ∈ Q…

A: Given: a and b belong to Q with b ≠ 0. To Prove: Qa=Qb if and only if there exists some c∈Q such…

Q: Consider f: A→B and let S, T ⊆ A. 1. Prove: f(S∩T)⊆f(S)∩f(T) 2. Draw a diagram that shows why this…

A: Hi! There are multiple questions. As per norms, we will be answering only one question. As nothing…

Q: Let A and B be non-empty and bounded subsets of R such that inf A < inf B. Prove the following: ∃a…

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Q: Let A and B be subsets of R which are bounded above. a. Show that sup(AUB) = max {sup(A), sup(B)}.…

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