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.

Expert Solution

Colby Messinger answered 2 years ago

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

Consider the function ?: ℝ → ℝ defined by ?(?) = ? if ? ∈ ℚ...

Consider the function ?: ℝ → ℝ defined by ?(?) = ? if ? ∈ ℚ and ?(?) = ? 2 if ? ∈ ℝ ∖ ℚ. Find all points at which ? is continuous

Consider the function ?: (−1,1) × (−1,1) → ℝ given by ?(?, ?) = sin(?? +...

Consider the function ?: (−1,1) × (−1,1) → ℝ given by ?(?, ?) = sin(?? + ?? + ?2 ). 1. Find a bound for the directional derivative of ? in any direction, i.e. find a constant ? such that |???(?, ?)| ≤ ? for all (?, ?) ∈ (−1,1) × (−1,1) and ? ∈ ℝ 2 with |?| = 1.

For each of the subspaces ? and ? of ?4 (ℝ) defined below, find bases for...

For each of the subspaces ? and ? of ?4 (ℝ) defined below, find bases for ? + ? and ? ∩ ?, and verify that dim[?] + dim[? ] = dim[? + ? ] + dim[? ∩ ? ] (a) ? = {(1, 3, 0, 1), (1, −2, 2, −2)}, ? = {(1, 0, −1, 0), (2, 1, 2, −1)} (b) ? = {(1, 0, 1, 2), (1, 1, 0, 1)}, ? = {(0, 2, 1, 1), (2, 0,...

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)

Topology question: Show that a function f : ℝ → ℝ is continuous in the ε...

Topology question: Show that a function f : ℝ → ℝ is continuous in the ε − δ definition of continuity if and only if, for every x ∈ ℝ and every open set U containing f(x), there exists a neighborhood V of x such that f(V) ⊂ U.

(1)Prove 6^(2n)-4^(2n) must be a mutiple of 20 (2)Prove 6^(2n)+4^(2n)-2 must be a multiple of 50

4.Consider events A and B, neither of which are impossible events. Prove the following statements. A)(4...

4.Consider events A and B, neither of which are impossible events. Prove the following statements. A)? ? ? = 1 − ?(?|?) B)If ? ? ? ? = ? ? ∩ ? , then A and B must be independent events.

Prove these scenarios by mathematical induction: (1) Prove n2 < 2n for all integers n>4 (2)...

Prove these scenarios by mathematical induction: (1) Prove n2 < 2n for all integers n>4 (2) Prove that a finite set with n elements has 2n subsets (3) Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps

Prove that R(4; 4) = 18.

Question