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 #
Real-Analysis
Surjective Function
A function is surjective or “onto” if For every y∈Y, there exists x∈X such that f(x)=y.
Peano Axioms
A natural number is any element of the set 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++ ≠ 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≠x′⟹f(x)≠f(x′) i.e. Given a set X and a set Y, no two elements (say, x and x′) 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×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×Y={(x,y)|x∈X,y∈Y} An example would be R×R=R2 (i.e. the cartesian plane). R2 is the cartesian product of “crossing” 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