Definition of Sets and Propositional Functions
Definitions 1
- Set: A collection of distinct objects that are the subjects of our intuition or thought is called a set.
- Element: An object that belongs to a set is called an element.
- 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
- 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.
- 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 $N \in \mathbb{N}$.
- 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.
- 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.
- 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.
Translated by Heung-Chun Lee, You-Feng Lin. (2011). Set Theory: An Intuitive Approach: p47, 73, 81, 85. ↩︎