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/- Copyright (c) Heather Macbeth, 2023-4. All rights reserved. -/ | ||
import Mathlib.Data.Real.Basic | ||
import Library.Basic | ||
import Library.Tactic.ModEq | ||
import AutograderLib | ||
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math2001_init | ||
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/-! # Homework 10 | ||
Don't forget to compare with the text version, | ||
https://github.com/hrmacbeth/math2001/wiki/Homework-10, | ||
for clearer statements and any special instructions. -/ | ||
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@[autograded 4] | ||
theorem problem1a : { m : ℤ | m ≥ 10 } ⊆ { n : ℤ | n ^ 3 - 8 * n ^ 2 ≥ 2 * n } := by | ||
sorry | ||
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@[autograded 4] | ||
theorem problem1b : { m : ℤ | m ≥ 10 } ⊈ { n : ℤ | n ^ 3 - 7 * n ^ 2 ≥ 4 * n } := by | ||
sorry | ||
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@[autograded 4] | ||
theorem problem2a : { t : ℝ | t ^ 2 - 5 * t + 4 = 0 } = { s : ℝ | s = 4 } := by | ||
sorry | ||
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@[autograded 4] | ||
theorem problem2b : { t : ℝ | t ^ 2 - 5 * t + 6 = 0 } ≠ { s : ℝ | s = 2 } := by | ||
sorry | ||
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@[autograded 4] | ||
theorem problem3a : {1, 2, 3} = {1, 2} := by | ||
sorry | ||
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@[autograded 4] | ||
theorem problem3b : {1, 2, 3} ≠ {1, 2} := by | ||
sorry | ||
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@[autograded 4] | ||
theorem problem4 : { r : ℤ | r ≡ 8 [ZMOD 10] } | ||
⊆ { s : ℤ | s ≡ 0 [ZMOD 2] } ∩ { t : ℤ | t ≡ 3 [ZMOD 5] } := by | ||
sorry | ||
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/-! ### Problem 5 starts here -/ | ||
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infix:50 "∼" => fun (x y : ℤ) ↦ x + y ≡ 0 [ZMOD 3] | ||
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@[autograded 2] | ||
theorem problem51a : Reflexive (· ∼ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem51b : ¬ Reflexive (· ∼ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem52a : Symmetric (· ∼ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem52b : ¬ Symmetric (· ∼ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem53a : AntiSymmetric (· ∼ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem53b : ¬ AntiSymmetric (· ∼ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem54a : Transitive (· ∼ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem54b : ¬ Transitive (· ∼ ·) := by | ||
sorry | ||
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/-! ### Problem 6 starts here -/ | ||
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infix:50 "≺" => fun ((x1, y1) : ℝ × ℝ) (x2, y2) ↦ (x1 ≤ x2 ∧ y1 ≤ y2) | ||
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@[autograded 2] | ||
theorem problem61a : Reflexive (· ≺ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem61b : ¬ Reflexive (· ≺ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem62a : Symmetric (· ≺ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem62b : ¬ Symmetric (· ≺ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem63a : AntiSymmetric (· ≺ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem63b : ¬ AntiSymmetric (· ≺ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem64a : Transitive (· ≺ ·) := by | ||
sorry | ||
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@[autograded 2] | ||
theorem problem64b : ¬ Transitive (· ≺ ·) := by | ||
sorry |