Fundamental Equations and definitions

Function notation

• $f:X\to Y$ means $f$ is a Function from domain $X$ to codomain $Y$.
• $x\mapsto f(x)$ describes the rule applied to an input.
• $\mathrm{dom}(f)=X$ is the Domain.
• $\mathrm{im}(f)=f(X)=\{f(x):x\in X\}$ is the image/range; see Codomain and range.
• $g\circ f$ means “apply $f$ first, then $g$”: $(g\circ f)(x)=g(f(x))$.

Inverses

A function $f:X\to Y$ has a two-sided inverse $f^{-1}:Y\to X$ when $f^{-1}(f(x))=x\text{for all }x\in X$ and $f(f^{-1}(y))=y\text{for all }y\in Y.$ Equivalently, $f$ must be bijective between the chosen domain and codomain.

Natural exponential

Euler’s number can be defined by

ight)^n=\sum_{n=0}^{\infty}\frac1{n!}.$$ The exponential $e^x$ and natural logarithm $\ln x$ are inverse functions on compatible domains: $$\ln(e^x)=x,\qquad e^{\ln x}=x\quad(x>0).$$ ## Physics reminder For a time-dependent position $s(t)$, the derivative $ds/dt$ is [[Velocity]]. The notation only becomes physically meaningful after the domain, variable, and units are clear. ## Equality versus definition Use equality when two expressions are the same value:

(a+b)^2=a^2+2ab+b^2.

$$Usedefinitionnotationwhenintroducingmeaning:$$

f(x) := x^2+1.

That tiny distinction helps in proofs and derivations: definitions are choices; equalities are consequences. ## Quantifiers Common proof notation: - $orall$ means "for all". - $\exists$ means "there exists". - $\exists!$ means "there exists exactly one". These show up constantly in [[Set theory index]] and function definitions.