1. Find the supremum, infimum, maximum, and minimum of each subset of R, if it exists. a. A = U[-¹, 1-¹) ΠΕΝ sup A max A inf A min A b. B = {D" :n € N} sup B max B inf B min B c. C = {r € Q:r² ≥ 1} sup C max C inf C min C

Q: Let F= R and V = R*. Let W, = {(x1,x2,X3,X4) € R* : x1 – x2 + X3 – x4 = 0 and x1 +x2 = 0}, W2 =…

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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: C. Show that both R and R² may be covered by countably many open balls.

A: Firstly we have shown that R may be covered by countably many open balls. Let us  consider the set…

Q: Suppose n E N and u, v € V₁ such that |u| = 5, |v|= 4 and u⚫v=-7 Find |3u + v. Find t R such that u…

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Q: For each of the following sets, find the maximum, minimum, supremum, and infimum IF THEY EXIST. S =…

A: When a set is bounded above, it has upper bounds. The least of all upper bounds is the supremum.…

Q: Let a, b, n E Z. Prove: If a|n and b|n with gcd(a, b) = 1, then ab|n. (I

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Q: Let x, y be two positive real numbers such that a?y6 + 4 = a where a > 4. What is the minimum value…

A: To find the minimum value for x+y3 if x2y6+4=a where a≥4.

Q: Find the infimums of the sets S={1/n : n∈ ℕ+} and T={ (-1)n/n : n∈ ℕ+}. Is either infimum a minimum.…

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Q: [2.28] Which of the following sets are convex and which are not? a. {(x₁, x₂) x² + x² ≥ 3}. : b.…

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Q: Prove that |A U B| + |A n B| = |A| + |B|

A: Let A and B are two sets. A = number of elements in the set A. B = number of elements in the set B.…

Q: 16. Of all numbers whose sum is 340, find the two that have the maximum product. A. 10 and 330 B.…

A: Let one number be x and the other number y, then                x+y= 340       ⇒  y=340-x…

Q: If X, Y, Z ER and x≤Z, show that XL Y S z if and only if |x-y| + | y-z| = |x-z). Interpret…

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Q: 2b. In R show that : || x - y || || x + y || ≤ 11x11 ² + 11y11²2² -(1

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Q: b. Min z = 4x1 + 2x2 Sute 3x1 + x2 2 27 -X1 - x2 <21 х1 + 2х; 2 30 xị and x; are unrestricted.

A: Step:-1 Given that Min z =4x1 +2x2s.t.3x1 +x2≥27-x1 -x2≤21x1 +2x2≥30x1, x2 are unrestricted Take…

Q: Let x ∈ R. Prove that if x 0.

A: Given :

Q: Q: State whether the following statements are true or false: 1. Let S be a non-empty subset of the…

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Q: The floor function is defined as bxc = max{m ∈ Z | m ≤ x}. Effectively, the floor function is the…

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