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Real Analysis Differential Geometry Linear Algebra Topology Set Theory General Concepts # Bijective Function # Injective Function # Surjective Function # Transitive Relation # Symmetric Relation # Locality # Peano Axioms #

Peano Axioms

Math Real-Analysis

A natural number is any element of the set \(\mathbb{N} = \{0, 1, 2, 3…\}\) I # 0 is a natural number. II # If \(n\) is a natural number, then n++ is also a natural number. III # 0 is not the successor of any natural number; i.e., we have n++ \(\ne\) 0 for every natural number n. IV # Different natural numbers must have different successors; i. ...

Injective Function

Math Set-Theory Real-Analysis

A function is injective (“one-to-one”) if \(x \neq x^{\prime} \Longrightarrow f(x) \neq f(x^{\prime})\) i.e. Given a set \(X\) and a set \(Y\), no two elements (say, \(x\) and \(x^{\prime}\)) from \(X\) map to the same element in \(Y\). (Note: a function can be both injective and surjective, this image is injective only)

Cartesian Product

Math Real-Analysis

If \(X\) and \(Y\) are sets, then we define the Cartesian Product \(X \times Y\) to be the collection of ordered pairs, e.g. \((x, y)\), whose first component lies in \(X\) and whose second component lies in \(Y\) \(X \times Y = \{(x, y) | x \in X, y \in Y\}\) An example would be \(\mathbb{R} \times \mathbb{R} = \mathbb{R}^{2}\) (i.e. the cartesian plane). \(\mathbb{R}^{2}\) is the cartesian product of “crossing” \(\mathbb{R}\) with itself. ...

Bijective Function

Math Set-Theory Real-Analysis

A function is bijective or “invertible” if it is Both one-to-one and onto (injective and surjective) In other words: each element of one set is paired with exactly one element of the other set, and vice versa