Update Mon Dec 9 12:09:20 EST 2024

Math2001/Homework/hw10.lean

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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
+
+math2001_init
+set_option quotPrecheck false
+
+
+/-! # 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. -/
+
+
+/- Problem 1: prove one of these, delete the other -/
+
+@[autograded 4]
+theorem problem1a : { m : ℤ | m ≥ 10 } ⊆ { n : ℤ | n ^ 3 - 6 * n ^ 2 ≥ 4 * n } := by
+  sorry
+
+@[autograded 4]
+theorem problem1b : { m : ℤ | m ≥ 10 } ⊈ { n : ℤ | n ^ 3 - 6 * n ^ 2 ≥ 4 * n } := by
+  sorry
+
+
+/- Problem 2: prove one of these, delete the other -/
+
+@[autograded 3]
+theorem problem2a : { t : ℝ | t ^ 2 - 3 * t + 2 = 0 } = { s : ℝ | s = 2 } := by
+  sorry
+
+@[autograded 3]
+theorem problem2b : { t : ℝ | t ^ 2 - 3 * t + 2 = 0 } ≠ { s : ℝ | s = 2 } := by
+  sorry
+
+
+/- Problem 3: prove one of these, delete the other -/
+
+@[autograded 3]
+theorem problem3a : {1, 2, 3} ∩ {2, 3, 4} ⊆ {2, 3, 6} := by
+  sorry
+
+
+@[autograded 3]
+theorem problem3b : {1, 2, 3} ∩ {2, 3, 4} ≠ {2, 3, 6} := by
+  sorry
+
+
+/- Problem 4 -/
+
+@[autograded 4]
+theorem problem4 : { r : ℤ | r ≡ 11 [ZMOD 15] }
+    = { s : ℤ | s ≡ 2 [ZMOD 3] } ∩ { t : ℤ | t ≡ 1 [ZMOD 5] } := by
+  sorry
+
+
+
+/-! ### Problem 5 starts here -/
+
+
+local infix:50 "∼" => fun (a b : ℤ) ↦ ∃ m n, m > 0 ∧ n > 0 ∧ a * m = b * n
+
+
+/- Problem 5.1: prove one of these, delete the other -/
+
+@[autograded 2]
+theorem problem51a : Reflexive (· ∼ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem51b : ¬ Reflexive (· ∼ ·) := by
+  sorry
+
+
+/- Problem 5.2: prove one of these, delete the other -/
+
+@[autograded 2]
+theorem problem52a : Symmetric (· ∼ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem52b : ¬ Symmetric (· ∼ ·) := by
+  sorry
+
+
+/- Problem 5.3: prove one of these, delete the other -/
+
+@[autograded 2]
+theorem problem53a : AntiSymmetric (· ∼ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem53b : ¬ AntiSymmetric (· ∼ ·) := by
+  sorry
+
+
+/- Problem 5.4: prove one of these, delete the other -/
+
+@[autograded 2]
+theorem problem54a : Transitive (· ∼ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem54b : ¬ Transitive (· ∼ ·) := by
+  sorry
+
+
+
+/-! ### Problem 6 starts here -/
+
+infix:50 "≺" => fun ((x1, y1) : ℝ × ℝ) (x2, y2) ↦ (x1 ≤ x2 ∧ y1 ≤ y2)
+
+
+/- Problem 6.1: prove one of these, delete the other -/
+
+@[autograded 2]
+theorem problem61a : Reflexive (· ≺ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem61b : ¬ Reflexive (· ≺ ·) := by
+  sorry
+
+
+/- Problem 6.2: prove one of these, delete the other -/
+
+@[autograded 2]
+theorem problem62a : Symmetric (· ≺ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem62b : ¬ Symmetric (· ≺ ·) := by
+  sorry
+
+
+/- Problem 6.3: prove one of these, delete the other -/
+
+@[autograded 2]
+theorem problem63a : AntiSymmetric (· ≺ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem63b : ¬ AntiSymmetric (· ≺ ·) := by
+  sorry
+
+
+/- Problem 6.4: prove one of these, delete the other -/
+
+@[autograded 2]
+theorem problem64a : Transitive (· ≺ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem64b : ¬ Transitive (· ≺ ·) := by
+  sorry