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1. Cardinality

   Linked via "bijection"

   Historical Development
   The modern rigorous concept of cardinality was formalized by Georg Cantor in the late 19th century. Cantor established that two sets have the same cardinality if and only if there exists a bijection (a one-to-one correspondence) between them. This fundamental equivalence allowed for the comparison of set sizes, even those that were apparently endless. Early challenges to this concept centered on the perceived inability of a proper subset to have the same "size" as t…

2. Cardinality

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   Finite Cardinality
   For any set $A$ containing a finite number of elements, its cardinality is simply the non-negative integer $n$ such that there is a bijection between $A$ and the set $\{1, 2, \ldots, n\}$. For the empty set $\emptyset$, the cardinality is 0.
   Countable Cardinality ($\aleph_0$)

3. Function Composition

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   Composition and Inverse Functions
   The relationship between composition and inverse functions is foundational to group theory. If a function $f: A \to B$ is a bijection (both injective and surjective), then its inverse, $f^{-1}: B \to A$, exists. The composition of a function with its inverse yields the [identity mapping](/entries/identity-mappi…

4. Symmetry Group

   Linked via "bijective functions"

   Formal Definition and Algebraic Structure
   A symmetry group, denoted $G$, is formally defined as the set of all bijective functions (or mappings) from a set $S$ onto itself, $g: S \to S$, such that for any element $s \in S$, the application of $g$ leaves some defined property $\mathcal{P}(s)$ unchanged.
   The set $G$ must satisfy the axioms of a group under the operation of functional composition $(\circ)$: