Enderton.Set.Chapter_1

Introduction

Exercise 1.1

Which of the following become true when "∈" is inserted in place of the blank? Which become true when "⊆" is inserted?

theorem Enderton.Set.Chapter_1.empty_ne_singleton_empty {α : Type u_1} (h : ∅ = {∅}) : False

The ∅ does not equal the singleton set containing ∅.

theorem Enderton.Set.Chapter_1.exercise_1_1a {α : Type u_1} : {∅} ∈ {∅, {∅}} ∧ {∅} ⊆ {∅, {∅}}

Exercise 1.1a

{∅} ___ {∅, {∅}}

theorem Enderton.Set.Chapter_1.exercise_1_1b {α : Type u_1} : {∅} ∉ {∅, {{∅}}} ∧ {∅} ⊆ {∅, {{∅}}}

Exercise 1.1b

{∅} ___ {∅, {{∅}}}

theorem Enderton.Set.Chapter_1.exercise_1_1c {α : Type u_1} : {{∅}} ∉ {∅, {∅}} ∧ {{∅}} ⊆ {∅, {∅}}

Exercise 1.1c

{{∅}} ___ {∅, {∅}}

theorem Enderton.Set.Chapter_1.exercise_1_1d {α : Type u_1} : {{∅}} ∈ {∅, {{∅}}} ∧ ¬{{∅}} ⊆ {∅, {{∅}}}

Exercise 1.1d

{{∅}} ___ {∅, {{∅}}}

theorem Enderton.Set.Chapter_1.exercise_1_1e {α : Type u_1} : {{∅}} ∉ {∅, {∅, {∅}}} ∧ ¬{{∅}} ⊆ {∅, {∅, {∅}}}

Exercise 1.1e

{{∅}} ___ {∅, {∅, {∅}}}

theorem Enderton.Set.Chapter_1.exercise_1_2 {α : Type u_1} : ∅ ≠ {∅} ∧ ∅ ≠ {{∅}} ∧ {∅} ≠ {{∅}}

Exercise 1.2

Show that no two of the three sets ∅, {∅}, and {{∅}} are equal to each other.

theorem Enderton.Set.Chapter_1.exercise_1_3 : ∀ {α : Type u_1} {B C : Set α}, B ⊆ C → 𝒫 B ⊆ 𝒫 C

Exercise 1.3

Show that if B ⊆ C, then 𝓟 B ⊆ 𝓟 C.

theorem Enderton.Set.Chapter_1.exercise_1_4 {α : Type u_1} {B : Set α} (x : α) (y : α) (hx : x ∈ B) (hy : y ∈ B) : {{x}, {x, y}} ∈ 𝒫 𝒫 B

Exercise 1.4

Assume that x and y are members of a set B. Show that {{x}, {x, y}} ∈ 𝓟 𝓟 B.