Draft up Exercises 1.11.

finite-set-exercises
Joshua Potter 2023-05-08 13:18:12 -06:00
parent 2ec4611309
commit df1537b71a
9 changed files with 263 additions and 15 deletions

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@ -1,6 +1,7 @@
import Bookshelf.Real.Basic
import Bookshelf.Real.Function
import Bookshelf.Real.Geometry
import Bookshelf.Real.Int
import Bookshelf.Real.Rational
import Bookshelf.Real.Sequence
import Bookshelf.Real.Set

10
Bookshelf/Real/Int.lean Normal file
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import Mathlib.Data.Real.Basic
namespace Real
/--
Check whether a real number is an integer.
-/
def isInt (x : ) := x = Int.floor x
end Real

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import Exercises.Apostol.Chapter_I_3
import Exercises.Apostol.Chapter_I_03
import Exercises.Apostol.Chapter_1_11

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import Bookshelf.Real.Int
import Mathlib.Data.Real.Basic
/-! # Exercises.Apostol.Exercises_1_11 -/
namespace Exercises.Apostol.Exercises_1_11
/-! ## Exercise 4
Prove that the greatest-integer function has the properties indicated.
-/
/-- ### Exercise 4a
`⌊x + n⌋ = ⌊x⌋ + n` for every integer `n`.
-/
theorem exercise_4a (x : ) (n : ) : ⌊x + n⌋ = ⌊x⌋ + n := by
sorry
/-- ### Exercise 4b
`⌊-x⌋ = -⌊x⌋` if `x` is an integer.
`⌊-x⌋ = -⌊x⌋ - 1` otherwise.
-/
theorem exercise_4b (x : )
: (Real.isInt x → ⌊-x⌋ = -⌊x⌋)
(¬Real.isInt x → ⌊-x⌋ = -⌊x⌋ - 1) := by
sorry
/-- ### Exercise 4c
`⌊x + y⌋ = ⌊x⌋ + ⌊y⌋` or `⌊x⌋ + ⌊y⌋ + 1`.
-/
theorem exercise_4c (x y : )
: ⌊x + y⌋ = ⌊x⌋ + ⌊y⌋ ⌊x + y⌋ = ⌊x⌋ + ⌊y⌋ + 1 := by
sorry
/-- ### Exercise 4d
`⌊2x⌋ = ⌊x⌋ + ⌊x + 1/2⌋`
-/
theorem exercise_4d (x : )
: ⌊2 * x⌋ = ⌊x⌋ + ⌊x + 1/2⌋ := by
sorry
/-- ### Exercise 4e
`⌊3x⌋ = ⌊x⌋ + ⌊x + 1/3⌋ + ⌊x + 2/3⌋`
-/
theorem exercise_4e (x : )
: ⌊3 * x⌋ = ⌊x⌋ + ⌊x + 1/3⌋ + ⌊x + 2/3⌋ := by
sorry
/-- ### Exercise 5
The formulas in Exercises 4(d) and 4(e) suggest a generalization for `⌊nx⌋`.
State and prove such a generalization.
-/
theorem exercise_5 (n : ) (x : )
: ⌊n * x⌋ = 10 := by
sorry
/-- ### Exercise 7b
If `a` and `b` are positive integers with no common factor, we have the formula
`Σ_{n=1}^{b-1} ⌊na / b⌋ = ((a - 1)(b - 1)) / 2`. When `b = 1`, the sum on the
left is understood to be `0`.
Derive the result analytically as follows: By changing the index of summation,
note that `Σ_{n=1}^{b-1} ⌊na / b⌋ = Σ_{n=1}^{b-1} ⌊a(b - n) / b⌋`. Now apply
Exercises 4(a) and (b) to the bracket on the right.
-/
theorem exercise_7b : True := sorry
end Exercises.Apostol.Exercises_1_11

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\documentclass{article}
\usepackage{amsmath}
\usepackage[shortlabels]{enumitem}
\input{../../preamble}
\newcommand{\link}[1]{\lean{../..}
{Exercises/Apostol/Exercises\_1\_11}
{Exercises.Apostol.Exercises\_1\_11.#1}
{Exercises\_1\_11.#1}
}
\begin{document}
\section{Exercise 4}%
\label{sec:exercise-4}
Prove that the greatest-integer function has the properties indicated:
\subsection{Exercise 4a}%
\label{sub:exercise-4a}
$\floor{x + n} = \floor{x} + n$ for every integer $n$.
\begin{proof}
\link{exercise\_4a}
\end{proof}
\subsection{Exercise 4b}%
\label{sub:exercise-4b}
$\floor{-x} =
\begin{cases}
-\floor{x} & \text{if } x \text{ is an integer}, \\
-\floor{x} - 1 & \text{otherwise}.
\end{cases}$
\begin{proof}
\link{exercise\_4b}
\end{proof}
\subsection{Exercise 4c}%
\label{sub:exercise-4c}
$\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
\begin{proof}
\link{exercise\_4c}
\end{proof}
\subsection{Exercise 4d}%
\label{sub:exercise-4d}
$\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
\begin{proof}
\link{exercise\_4d}
\end{proof}
\subsection{Exercise 4e}%
\label{sub:exercise-4e}
$\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
\begin{proof}
\link{exercise\_4e}
\end{proof}
\section{Exercise 5}%
\label{sec:exercise-5}
The formulas in Exercises 4(d) and 4(e) suggest a generalization for
$\floor{nx}$.
State and prove such a generalization.
\begin{proof}
\link{exercise\_5}
\end{proof}
\section{Exercise 6}%
\label{sec:exercise-6}
Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are
integers.
Let $f$ be a nonnegative function whose domain is the interval $[a, b]$, where
$a$ and $b$ are integers, $a < b$.
Let $S$ denote the set of points $(x, y)$ satisfying $a \leq x \leq b$,
$0 < y \leq f(x)$.
Prove that the number of lattice points in $S$ is equal to the sum
$$\sum_{n=a}^b \floor{f(n)}.$$
\begin{proof}
TODO
\end{proof}
\section{Exercise 7}%
\label{sec:exercise-7}
If $a$ and $b$ are positive integers with no common factor, we have the formula
$$\sum_{n=1}^{b-1} \floor{\frac{na}{b}} = \frac{(a - 1)(b - 1)}{2}.$$
When $b = 1$, the sum on the left is understood to be $0$.
\subsection{Exercise 7a}%
\label{sub:exercise-7a}
Derive this result by a geometric argument, counting lattice points in a right
triangle.
\begin{proof}
TODO
\end{proof}
\subsection{Exercise 7b}%
\label{sub:exercise-7b}
Derive the result analytically as follows:
By changing the index of summation, note that
$\sum_{n=1}^{b-1} \floor{na / b} = \sum_{n=1}^{b-1} \floor{a(b - n) / b}$.
Now apply Exercises 4(a) and (b) to the bracket on the right.
\begin{proof}
TODO
\end{proof}
\section{Exercise 8}%
\label{sec:exercise-8}
Let $S$ be a set of points on the real line.
The \textit{characteristic function} of $S$ is, by definition, the function
$\chi_S$ such that $\chi_S(x) = 1$ for every $x$ in $S$, and $\chi_S(x) = 0$
for those $x$ not in $S$.
Let $f$ be a step function which takes the constant value $c_k$ on the $k$th
open subinterval $I_k$ of some partition of an interval $[a, b]$.
Prove that for each $x$ in the union $I_1 \cup I_2 \cup \cdots \cup I_n$ we have
$$f(x) = \sum_{k=1}^n c_k\chi_{I_k}(x).$$
This property is described by saying that every step function is a linear
combination of characteristic functions of intervals.
\begin{proof}
TODO
\end{proof}
\end{document}

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@ -10,8 +10,6 @@ namespace Exercises.Apostol.Chapter_I_3
#check Archimedean
#check Real.exists_isLUB
namespace Real
/-! ## The least-upper-bound axiom (completeness axiom) -/
/--
@ -464,7 +462,7 @@ theorem forall_mem_le_forall_mem_imp_sup_le_inf (S T : Set )
intro x hx
exact p s hs x hx
have ⟨S_lub, hS_lub⟩ := Real.exists_isLUB S hS ⟨t, ps⟩
have ⟨T_glb, hT_glb⟩ := Real.exists_isGLB T hT ⟨s, pt⟩
have ⟨T_glb, hT_glb⟩ := exists_isGLB T hT ⟨s, pt⟩
refine ⟨S_lub, ⟨hS_lub, ⟨T_glb, ⟨hT_glb, ?_⟩⟩⟩⟩
-- Assume `T_glb < S_lub`. Then `∃ c, T_glb + c < S_lub` which in turn implies
-- existence of some `x ∈ S` such that `T_glb < S_lub - c / 2 < x < S_lub`.
@ -489,8 +487,6 @@ theorem forall_mem_le_forall_mem_imp_sup_le_inf (S T : Set )
_ < x := hx.right
simp at this
end Real
/-! ## Exercises -/
/-- #### Exercise 1
@ -498,7 +494,7 @@ end Real
If `x` and `y` are arbitrary real numbers with `x < y`, prove that there is at
least one real `z` satisfying `x < z < y`.
-/
theorem exercise1 (x y : ) (h : x < y) : ∃ z, x < z ∧ z < y := by
theorem exercise_1 (x y : ) (h : x < y) : ∃ z, x < z ∧ z < y := by
have ⟨z, hz⟩ := exists_pos_add_of_lt' h
refine ⟨x + z / 2, ⟨?_, ?_⟩⟩
· have hz' : z / 2 > 0 := by
@ -515,15 +511,15 @@ theorem exercise1 (x y : ) (h : x < y) : ∃ z, x < z ∧ z < y := by
If `x` is an arbitrary real number, prove that there are integers `m` and `n`
such that `m < x < n`.
-/
theorem exercise2 (x : ) : ∃ m n : , m < x ∧ x < n := by
theorem exercise_2 (x : ) : ∃ m n : , m < x ∧ x < n := by
refine ⟨x - 1, ⟨x + 1, ⟨?_, ?_⟩⟩⟩ <;> norm_num
/-- #### Exercise 3
If `x > 0`, prove that there is a positive integer `n` such that `1 / n < x`.
-/
theorem exercise3 (x : ) (h : x > 0) : ∃ n : +, 1 / n < x := by
have ⟨n, hn⟩ := @Real.exists_pnat_mul_self_geq_of_pos x 1 h
theorem exercise_3 (x : ) (h : x > 0) : ∃ n : +, 1 / n < x := by
have ⟨n, hn⟩ := @exists_pnat_mul_self_geq_of_pos x 1 h
refine ⟨n, ?_⟩
have hr := mul_lt_mul_of_pos_right hn (show 0 < 1 / ↑↑n by norm_num)
conv at hr => arg 2; rw [mul_comm, ← mul_assoc]; simp
@ -536,7 +532,7 @@ which satisfies the inequalities `n ≤ x < n + 1`. This `n` is called the
greatest integer in `x` and is denoted by `⌊x⌋`. For example, `⌊5⌋ = 5`,
`⌊5 / 2⌋ = 2`, `⌊-8/3⌋ = -3`.
-/
theorem exercise4 (x : ) : ∃! n : , n ≤ x ∧ x < n + 1 := by
theorem exercise_4 (x : ) : ∃! n : , n ≤ x ∧ x < n + 1 := by
let n := Int.floor x
refine ⟨n, ⟨?_, ?_⟩⟩
· exact ⟨Int.floor_le x, Int.lt_floor_add_one x⟩
@ -549,7 +545,7 @@ theorem exercise4 (x : ) : ∃! n : , n ≤ x ∧ x < n + 1 := by
If `x` is an arbitrary real number, prove that there is exactly one integer `n`
which satisfies `x ≤ n < x + 1`.
-/
theorem exercise5 (x : ) : ∃! n : , x ≤ n ∧ n < x + 1 := by
theorem exercise_5 (x : ) : ∃! n : , x ≤ n ∧ n < x + 1 := by
let n := Int.ceil x
refine ⟨n, ⟨?_, ?_⟩⟩
· exact ⟨Int.le_ceil x, Int.ceil_lt_add_one x⟩

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\input{../../preamble}
\newcommand{\link}[1]{\lean{../..}
{Exercises/Apostol/Chapter\_I\_3}
{Exercises.Apostol.Chapter\_I\_3.Real.#1}
{Chapter\_I\_3.Real.#1}
{Bookshelf/Apostol/Chapter\_I\_3}
{Apostol.Chapter\_I\_3.#1}
{Chapter\_I\_3.#1}
}
\begin{document}

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\hypersetup{colorlinks=true, urlcolor=blue}
\newcommand{\ceil}[1]{\left\lceil#1\right\rceil}
\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
% The first argument refers to a relative path upward from a current file to
% the root of the workspace (i.e. where this `preamble.tex` file is located).
\newcommand{\lean}[4]{\href{#1/#2.html\##3}{#4}}