📂Set Theory

Definition of Sets and Propositional Functions

Definitions ¹

1. Set: A collection of distinct objects that are the subjects of our intuition or thought is called a set.
2. Element: An object that belongs to a set is called an element.
3. Propositional Function: For an element $x$ of the set $U$, a proposition $p(x)$ that is either true or false is called a propositional function in $U$.

Explanation

1. In mathematics, the concept of a set is as important as nearly a native language. It might even be better than natural language because it eliminates ambiguity that necessarily follows and allows for logic to be developed through its definition and form alone.
2. Usually, elements are denoted by lowercase letters, and sets by uppercase letters. If $a$ belongs to $A$, it is denoted as $a \in A$ and said that $a$ is an element of $A$. Of course, it is not mandatory to represent elements and sets with uppercase and lowercase letters, respectively. The set of all natural numbers is commonly represented as $\mathbb{N}$, and there is no problem in denoting it as $N \in \mathbb{N}$.
   1. Enumeration: The set of natural numbers $\mathbb{N}$ can be represented as $\left\{ 1 , 2, 3, \cdots \right\}$. This method of directly writing the elements of a set is called the roster method.
   2. Set-Builder Notation: Unlike the roster method, sets can also be expressed as collections containing only those elements that satisfy a certain condition. For example, if you want to represent the set containing only natural numbers larger than $5$, it can be denoted as for a propositional function $p(x): x > 5$ by $\left\{ x \in \mathbb{N} : p(x) \text{ is truth} \right\}$. This can be more simply denoted as $\left\{ x \in \mathbb{N} : x > 5 \right\}$ without separately defining the propositional function. This notation is called set-builder notation.
3. It is important to note that a propositional function is defined as the propositional function itself. Although it conforms to the definition of a function in set theory, it is important that it can be defined solely by propositions. If this is unclear, the use of set-builder notation might become challenging due to the potential for functions to be circularly defined. Meanwhile, a propositional function is also called a logical formula.