Update Sat Apr 27 18:27:43 EDT 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
+
+
+/-! # 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. -/
+
+
+@[autograded 4]
+theorem problem1a : { m : ℤ | m ≥ 10 } ⊆ { n : ℤ | n ^ 3 - 8 * n ^ 2 ≥ 2 * n } := by
+  sorry
+
+@[autograded 4]
+theorem problem1b : { m : ℤ | m ≥ 10 } ⊈ { n : ℤ | n ^ 3 - 7 * n ^ 2 ≥ 4 * n } := by
+  sorry
+
+
+@[autograded 4]
+theorem problem2a : { t : ℝ | t ^ 2 - 5 * t + 4 = 0 } = { s : ℝ | s = 4 } := by
+  sorry
+
+@[autograded 4]
+theorem problem2b : { t : ℝ | t ^ 2 - 5 * t + 6 = 0 } ≠ { s : ℝ | s = 2 } := by
+  sorry
+
+
+@[autograded 4]
+theorem problem3a : {1, 2, 3} = {1, 2} := by
+  sorry
+
+@[autograded 4]
+theorem problem3b : {1, 2, 3} ≠ {1, 2} := by
+  sorry
+
+
+@[autograded 4]
+theorem problem4 : { r : ℤ | r ≡ 8 [ZMOD 10] }
+    ⊆ { s : ℤ | s ≡ 0 [ZMOD 2] } ∩ { t : ℤ | t ≡ 3 [ZMOD 5] } := by
+  sorry
+
+
+/-! ### Problem 5 starts here -/
+
+infix:50 "∼" => fun (x y : ℤ) ↦ x + y ≡ 0 [ZMOD 3]
+
+@[autograded 2]
+theorem problem51a : Reflexive (· ∼ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem51b : ¬ Reflexive (· ∼ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem52a : Symmetric (· ∼ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem52b : ¬ Symmetric (· ∼ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem53a : AntiSymmetric (· ∼ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem53b : ¬ AntiSymmetric (· ∼ ·) := by
+  sorry
+
+@[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)
+
+@[autograded 2]
+theorem problem61a : Reflexive (· ≺ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem61b : ¬ Reflexive (· ≺ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem62a : Symmetric (· ≺ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem62b : ¬ Symmetric (· ≺ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem63a : AntiSymmetric (· ≺ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem63b : ¬ AntiSymmetric (· ≺ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem64a : Transitive (· ≺ ·) := by
+  sorry
+
+@[autograded 2]
+theorem problem64b : ¬ Transitive (· ≺ ·) := by
+  sorry