📂Set Theory

Extensionality Axiom

Axiom ¹

$$\forall A \forall B ( \forall x ( x \in A \iff x \in B) )$$
Two sets $A$, $B$ are said to be equal if every element of the first is an element of the second and vice versa. This is represented as $A = B$.

Explanation

If $A$ and $B$ are not equal, it is denoted as $A \ne B$.

The equality of two sets is an axiom and a definition in itself. Extensionality refers not to extension but to external manifestation, meaning that a set is distinguished solely by the external relations of its elements, not by ambiguous explanations like ‘some kind of set.’ Considering the concept of elements as ‘objects distinctly distinguishable in our intuition or thought,’ this approach is valid.

In simpler terms, the axiom of extensionality ignores ’the essence’ of elements. Intuitively, both $a$ and $A$ appear to be the same letter, differing only in case. Regardless, if $a=A$ is stated, they are the same, and if $a \ne A$ is stated, they are seen as different.

According to this definition of ’equality,’ sets’ elements have no order or repetition. For example, the following equalities hold: $$\left\{ 9, 6, 0, 1, 2, 5 \right\} = \left\{ 0, 1, 2, 5, 6 ,9 \right\}$$

$$\left\{ y, y, x, y \right\} = \left\{ y , x \right\}$$

$$\left\{ 1, 5, 0, 4, 2, 1 \right\} = \left\{ 0, 1, 2, 4, 5 \right\}$$

In conjunction with inclusion relations of sets, the following useful theorem is introduced. This property is widely used throughout mathematics, particularly in pure mathematics fields such as abstract algebra or topology, where its application is fundamental. The axiom of extensionality is so intuitive that it’s sometimes underestimated, but it’s essential to recognize its fundamental importance.

Theorem

$$A = B \iff A \subset B \land B \subset A$$

Proof

$$\begin{align*} A = B \iff & \forall x ( (x \in A) \iff (x \in B)) \\ \iff & \forall x \left( ( x \in A \implies x \in B) \land ( x \in B \implies x \in A) \right) \\ \iff &A \subset B \land B \subset A \end{align*}$$

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