Let ?+?+?=7 where ?,?,?∈ℝ. Prove that ?^2+?^2+?^2≥7/3

Expert Solution

Colby Messinger answered 1 year ago

4) Consider ? ⊆ ℝ × ℝ with {(?,?)|?2 = ?2}. Prove that ? is an...

4) Consider ? ⊆ ℝ × ℝ with {(?,?)|?2 = ?2}. Prove that ? is an equivalence relation, and concisely characterize how its equivalence classes are different from simple real-number equality.

1A) Let ?(?) = 3? + 2. Use the ? − ? definition to prove that...

1A) Let ?(?) = 3? + 2. Use the ? − ? definition to prove that lim?→1 3? + 2 ≠ 1. Definition and proof. 1B) Let ?(?) = 2?^2 − 4? + 5. Use the ? − ? definition to prove that lim?→−1 2?^2 − 4? + 5 ≠ 8. definition and ? − ? Proof.

Define a function ?∶ ℝ→ℝ by ?(?)={?+1,[?] ?? ??? ?−1,[?]?? ???? where [x] is the integer...

Define a function ?∶ ℝ→ℝ by ?(?)={?+1,[?] ?? ??? ?−1,[?]?? ???? where [x] is the integer part function. Is ? injective? (b) Verify if the following function is bijective. If it is bijective, determine its inverse. ?∶ ℝ\{5/4}→ℝ\{9/4} , ?(?)=(9∙?)/(4∙?−5)

4) Let ? = {2, 3, 5, 7}, ? = {3, 5, 7}, ? = {1,...

4) Let ? = {2, 3, 5, 7}, ? = {3, 5, 7}, ? = {1, 7}. Answer the following questions, giving reasons for your answers. a) Is ? ⊆ ?? b)Is ? ⊆ ?? c) Is ? ⊂ ?? d) Is ? ⊆ ?? e) Is ? ⊆ ?? 5) Let ? = {1, 3, 4} and ? = {2, 3, 6}. Use set-roster notation to write each of the following sets, and indicate the number of elements in...

Let G be a group of order 42 = 2 * 3 * 7 (a) Let...

Let G be a group of order 42 = 2 * 3 * 7 (a) Let P7 be a Sylow 7-subgroup of G and let P3 be a Sylow 3-subgroup of G . What are the orders of P3 and P7? (b) Prove that P7 is the unique Sylow 7-subgroup of G and that P7 is normal. (c) Prove that P3P7 is a subgroup of G (d) Prove that P3P7 is a normal subgroup of G . (e) Let P2...

6. Let A = {1, 2, 3, 4} and B = {5, 6, 7}. Let f...

6. Let A = {1, 2, 3, 4} and B = {5, 6, 7}. Let f = {(1, 5),(2, 5),(3, 6),(x, y)} where x ∈ A and y ∈ B are to be determined by you. (a) In how many ways can you pick x ∈ A and y ∈ B such that f is not a function? (b) In how many ways can you pick x ∈ A and y ∈ B such that f : A → B...

7. Let m be a fixed positive integer. (a) Prove that no two among the integers...

7. Let m be a fixed positive integer. (a) Prove that no two among the integers 0, 1, 2, . . . , m − 1 are congruent to each other modulo m. (b) Prove that every integer is congruent modulo m to one of 0, 1, 2, . . . , m − 1.

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 +...

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 + · · · + n)^2 for every n ∈ N. That is, the sum of the first n perfect cubes is the square of the sum of the first n natural numbers. (As a student, I found it very surprising that the sum of the first n perfect cubes was always a perfect square at all.)

let G = D2n = {e,r,r^2,...,rn-1,s,sr,sr2,..,srn-1} a diedergroup of order 2n, where n >=3 (a) prove...

let G = D2n = {e,r,r^2,...,rn-1,s,sr,sr2,..,srn-1} a diedergroup of order 2n, where n >=3 (a) prove that [G,G] = <r2> (b) prove that G/[G,G] consists of two elements if n is uneven and 4 elements if n is even

Let S = {1, 2, 3, 4, 5, 6, 7} be a sample of an experiment...

Let S = {1, 2, 3, 4, 5, 6, 7} be a sample of an experiment and let X = {1, 4, 7}, Y = {2, 3, 5}, and Z = {1, 3, 5} be events. Which of the following statements is correct? a) X and S are mutually exclusive events. b) X and Y are mutually exclusive events. c) X, Y, and Z are mutually exclusive events. d) Z and Y are mutually exclusive events. e) X and Z...

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