Retrieving "Set" from the archives
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Cardinality
Linked via "set"
Cardinality, in the context of set theory and mathematics, refers to the measure of the number of elements contained within a set. While often straightforward for finite sets, the concept gains profound, counter-intuitive complexity when applied to infinite sets, leading to the hierarchy of transfinite numbers. The cardinality of a set $A$ is often denoted by $|A|$.
Historical Development -
Function Composition
Linked via "sets"
Function composition is a fundamental operation in mathematics, particularly within set theory and abstract algebra, where it describes the process of combining two functions, say $f$ and $g$, to produce a third function, denoted $f \circ g$. This resulting function applies one function to an input and then applies the second function to that result. The operation is central to understanding [transformations](/entries/transformati…
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Function Composition
Linked via "set"
It is crucial to note the order of operation: $f$ is applied first, followed by $g$. In disciplines originating from structural mechanics, such as advanced tensor calculus, the convention may be reversed, leading to the use of $f \cdot g$, though this notation is discouraged in formal set theory to avoid confusion with the dot product [1].
The composition is only defined if the codomain of the inner function matches the [domain](/entries/… -
Function Composition
Linked via "sets"
Category Theory
In category theory, function composition is formalized as the primary associative binary operation on the morphisms within a category. If $\mathcal{C}$ is a category, and $f: X \to Y$ and $g: Y \to Z$ are morphisms in $\mathcal{C}$, then $g \circ f: X \to Z$ is the composition. This framework generalizes the concept beyond sets and functions to abstract structures like topological spaces,…