📂Set Theory

Classification Axiomatic Form

Axioms ¹

$$ \forall X \exists A \forall a \left( a \in A \iff ( a \in X \land p(a)) \right) $$ For any set $X$, there exists a subset $A$ composed of elements that have property $p$.

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• $p(x)$ is a propositional function in $X$.

Explanation

The reason why $A$ is limited to a subset of $X$ is to prevent problems like Russell’s paradox. The reason it is not called an axiom but an axiom schema is that there are infinitely many such axioms, depending on the infinitely many $p(x)$. If there are two different propositional functions $p_{1}(x)$ and $p_{2}(x)$, what guarantees the existence of $\left\{ a \in X : p_{2}(a) \text{ is truth} \right\} \subset X$ is not the ‘Classification Axiom for $p_{1}(x)$’ but the ‘Classification Axiom for $p_{2}(x)$’.

Definition of Intersection and Difference Sets ²

The axiom schema of classification guarantees the existence of an intersection as defined below.

$$ x \in A \land x \in B \iff x \in A \cap B $$ For any two sets $A$, $B$, the set of elements belonging to both is called the intersection of $A$ and $B$, denoted as $A \cap B$.

Regarding the set $A$, the given propositional function is $p(x): x \in B$, which can be specifically written as $A \cap B= \left\{ x \in A : x \in B \right\}$. If $A \cap B = \emptyset$, then $A$ and $B$ are called disjoint.

Of course, the axiom schema of classification not only guarantees the existence of intersections but also the existence of all subsets satisfying a specific condition. This can be seen as the method of specifying conditions itself, one way of expressing sets.

$$ x \in A \land x \notin B \iff x \in A \setminus B $$ For any two sets $A$, $B$, the set of elements that belong to $A$ but not to $B$ is called the difference set of $A$ with respect to $B$, denoted as $A \setminus B$.

The set $U$ is called the complement of $A$, denoted as $A^{c}$. In considering the complement, the set $U$ is also referred to as the universal set.

Although set theory is infinite, not all branches of mathematics need to explore the entire abstract world. Typically, a certain universal set is predetermined as needed, and areas such as topology often use these concepts. Here are a few properties of complements and universal sets.

Basic Properties

Let’s assume sets $A$, $B$ are any subsets of the universal set $U$.

• [1] $$ \left(A^{c} \right)^{c} = A $$
• [2] $$ \emptyset^{c} = U \\ U^{c} = \emptyset $$
• [3] $$ A \cap A^{c} = \emptyset \\ A \cup A^{c} = U $$
• [4] $$ A \subset B \implies B^{c} \subset A^{c} $$
• [5] $$ A \setminus B = A \cap B^{c} $$

Proof

[5]

$$ \begin{align*} x \in A \setminus B &\iff x \in A \text{ and } x \notin B \\ &\iff x \in A \text{ and } x \in B^{c} \\ &\iff x \in A \cap B^{c} \end{align*} $$

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