Real-Analysis

## Math

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

## Surjective Function

A function is surjective or “onto” if For every $$y \in Y$$, there exists $$x \in X$$ such that $$f(x) = y$$.

## Peano Axioms

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

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

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

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