Sets

Introduction

In general, a set is a collection of objects. However, mathematically, a set is a mathematical model for representing a well-defined unordered collection of distinct objects. For example,

The collection of numbers, 9, 1, 5, 7, 3, 1, 3, can be viewed as a set that contains the five distinct elements 1, 3, 5, 7, 9.

A set is usually denoted by a capital letter, such as 𝑨, 𝑩, 𝑪, …, 𝑿, 𝒀, 𝒁, while lower-case letters, such as 𝑎, 𝑏, 𝑐, …, 𝑥, 𝑦, 𝑧, are used as symbols to denote unspecified objects of sets for a statement. For example,

Let 𝑨 be the set that contains the five distinct elements 1, 3, 5, 7, 9.
Let 𝑨 be the set of all elements 𝑎 is an odd natural numbers and 𝑎 is less than 10.

In other words, numbers, 1, 3, 5, 7, 9 are elements or members of set 𝑨. The term "is element of" is equivalent to "belongs to" or "is in". And the statement can be written as

1, 3, 5, 7, 9 ∈ 𝑨

While numbers 2, 4, 6, 8 are not elements or members of the set 𝑨. The term "not element of" is equivalent to "not belongs to" or "not in". And the statement can be written as

2, 4, 6, 8 ∉ 𝑨

The typical ways to define a set are roster form, semantic description and set-builder form.

Roster form defines a set by specifing the elements of a set as a comma-separated list, enclosed in curly brackets. For example, a set with a few elements
```
𝑨 = {1, 3, 5, 7, 9}
```
The roster form can also be used to define a set with many or infinite elements by abbreviating the ordered list of elements with an ellipsis, …. For example,
```
𝑩 = {1, 2, 3, …, 10000}
𝑪 = {1, 2, 3, …}
𝑫 = {…, -3, -2, -1, 0, 1, 2, 3, …}
```
Semantic description defines a set by specifying a well-defined rule or statement to determine the elements of the set. For example, Let 𝑨 be the set of all odd natural numbers less than 10. Let ℕ be the set of all natural numbers. ℕ = {1, 2, 3, …}, but sometimes ℕ may contain number 0. Let ℤ be the set of all integers. ℤ = {…, −2, −1, 0, 1, 2, …}. Let ℚ be the set of all rational numbers. Let ℝ be the set of all real numbers. Let ℂ be the set of all complex numbers. Let ℙ be the set of all prime numbers. ℙ = {2, 3, 5, 7, 11, 13, 17, 19, …}.
Set builder form defines a set by specifying the required properties of its member. For example, 𝑨={𝑥:𝑥∈ℕ and 𝑥 mod 2=1 and 𝑥<10} 𝑨={𝑥:𝑥 is an odd natural number less than 10} ℚ = {𝑎/𝑏 : 𝑎, 𝑏 ∈ ℤ, 𝑏 ≠ 0}. ℂ = {𝑎 + 𝑏𝑖 : 𝑎, 𝑏 ∈ ℝ}. In other words, 𝑨 is equal to the set of all elements 𝑥 such that 𝑥 is an odd natural number and 𝑥 is less than 10. ℚ is the set of all rational numbers. ℂ is the set of all complex numbers.

Special Sets of Numbers

Some special sets of numbers are

Let ℕ be the set of all natural numbers. ℕ = {1, 2, 3, …}, but sometimes ℕ may contain number 0.
Let ℕ^* be the set of all natural numbers without number 0.
    ℕ⁺ = ℕ₁ = ℕ^* = {1, 2, 3, …}.
    ℕ_>0 = {𝑛 ∈ ℕ : 𝑛 > 0}.
Let ℕ⁰ be the set of all natural numbers with number 0.
    ℕ₀ = ℕ⁰ = {0, 1, 2, 3, …}.
    ℕ_≥0 = {𝑛 ∈ ℕ : 𝑛 ≥ 0}.
Let ℤ be the set of all integers. ℤ = {…, −2, −1, 0, 1, 2, …}.
Let ℤ^* be the set of all integers without number 0.
    ℤ^* = {𝑛 ∈ ℤ : 𝑛 ≠ 0} = {…, −2, −1, 1, 2, …}
    (MathWorld: Let ℤ^* be the set of all non negative integers.)
Let ℤ⁺ be the set of all positive integers.
    ℤ⁺ = {1, 2, …} = ℕ_>0.
    ℤ_>0 = {𝑛 ∈ ℤ : 𝑛 > 0}.
Let ℤ^- be the set of all negative integers. 
    ℤ^- = {…, −2, −1}.
    ℤ_<0 = {𝑛 ∈ ℤ : 𝑛 < 0}
Let ℤ_≥0 be the set of all non-negative integers.
    ℤ_≥0 = {𝑛 ∈ ℤ : 𝑛 ≥ 0} = ℕ_≥0 = the set of whole numbers.
Let ℤ_≤0 be the set of all non-positive integers.
    ℤ_≤0 = {𝑛 ∈ ℤ : 𝑛 ≤ 0}.
For 𝑛 in ℕ let ℤ_𝑛 be the set of natural number less than 𝑛 with number 0.
    ℤ_𝑛 = {0, 1, 2, 𝑛-1}.
For 𝑛 in ℕ let ℤ^⦻_n be the set of natural number less than 𝑛 without number 0.
    ℤ_𝑛 = {1, 2, 𝑛-1}.

Let ℚ be the set of all rational numbers. ℚ = {𝑎/𝑏 : 𝑎, 𝑏 ∈ ℤ, 𝑏 ≠ 0}.
Let ℚ^* be the set of all rational numbers without number 0.
    ℚ^* = {𝑟 ∈ ℚ : 𝑟 ≠ 0} = {…, −2, −1, 1, 2, …}
ℚ_<0 = {𝑟 ∈ ℚ : 𝑟 < 0}.
Let ℝ be the set of all real numbers.
Let ℝ^* be the set of all real numbers without number 0.
    ℝ^* = {𝑥 ∈ ℝ : 𝑥 ≠ 0}
Let ℝ⁺ be the set of all positive real numbers.
    ℝ⁺ = {1, 2, …} = ℝ_>0.
    ℝ_>0 = {𝑥 ∈ ℝ : 𝑥 > 0}.
Let ℝ^- be the set of all negative real numbers. 
    ℝ^- = {…, −2, −1}.
    ℝ_<0 = {𝑥 ∈ ℝ : 𝑥 < 0}
Let ℝ_≥0 be the set of all non-negative real numbers.
    ℝ_≥0 = {𝑥 ∈ ℝ : 𝑥 ≥ 0}.
Let ℝ_≤0 be the set of all non-positive real numbers.
    ℝ_≤0 = {𝑥 ∈ ℝ : 𝑥 ≤ 0}.

Let ℂ be the set of all complex numbers. ℂ = {𝑎 + 𝑏𝑖 : 𝑎, 𝑏 ∈ ℝ}.
Let ℂ^* be the set of all complex numbers without number 0.
    ℂ^* = {𝑧 ∈ ℂ : 𝑧 ≠ 0}
Let ℙ be the set of all prime numbers. ℙ = {2, 3, 5, 7, 11, 13, 17, 19, …}.

Useful Logic

Logic is a way of thinking by inferring with the concept of correct reasoning. In other words, an output, either making a guess or forming an opinion, is produced based on using the speicified information as input. In general, logic is used to distinguish sound and faulty reasoning.

Propositions and Sentential Connectives

In order to analyse a problem logically, a precise language must be used.

Propositions

The precise building block used in logic reasoning is called proposition. A proposition is a declarative statement which is either true or false, but not both.

Sentential Connectives

In order to express mathematical statements to symbolic logical forms, some sentential connectives or logical connectives are defined. Keywords: 'not', 'and', 'or', 'if …, then …', 'if and only if' are the sentential connectives used in sentential logic. The logical connectives of sentential logic are Logic ConnectivesSentential ConnectivesSentential Logic SymbolRemarks NegativeNot¬∼, ! ConjunctionAnd∧., & DisjunctionInclusive Or∨+, ∥ Implication, Conditionalif …, then …→⇒ Double Implication, Biconditional, Equivalenceif and only if↔⇔, ≡

Arguments

Axiom of Extensionality

Basically, the foundation of set theory is based on the concept of belonging. In other words, a set is only used to represent the unordered elements contained in the set as in roster form. Both semantic description and set builder forms are only used to specify the elements in the set. However, rules used to determine the memners of the set may also be important properties of the elements of the set. Therefore, if the members of set 𝑨 is the same as the members of set 𝑩, set 𝑨 and set 𝑩 are equal. The equality of two sets A and B can be denoted as