Function

[!note] Definition Given sets $X$ and $Y$, a function from $X$ to $Y$ is a subset of $X \times Y$ which contains exactly one ordered pair $(x, y)$ for each $x \in X$

$\text{dom}R = \{x|x \in A, \exists y \in B, (x, y) \in R\}$ $\text{im}R = \text{range}R = \{y|y \in B, \exists x \in A, (x, y) \in R\}$ $\text{fld}R = \text{dom}R \cup \text{ran}R$

[!tip] Mappings Injective: one to one Surjective: onto OR correspondence Bijective: Injective + Surjective

[!question] Function Composition 定义: 如果有两个函数 $f: x \to y$ 和 $g:Y \to Z$，它们的复合函数记作 $g \circ f$ ，是从 $X$ 到 $Z$ 的一个新函数 。 If $f$ and $g$ are injective, then function composition $g \circ f$ is also injective. If $f$ and $g$ are surjective, then function composition $g \circ f$ is also surjective.

Inverse Functions

a) Identity Function

Definition: For any set $X$, its identity function $\iota_X$ (or $id_X$) is a function from $X$ to $X$, defined as $\iota_X(x) = x$. It’s a “do-nothing” function: what you put in is what you get out.

b) Definition of an Inverse Function

Definition: The inverse of a function $f:X \to Y$ (if it exists) is a function $g:Y \to X$ that satisfies the following two conditions:

1. $g \circ f = \iota_X$ (i.e., $g(f(x)) = x$ for all $x \in X$).
2. $f \circ g = \iota_Y$ (i.e., $f(g(y)) = y$ for all $y \in Y$).

Notation: The inverse of $f$ is usually denoted as $f^{-1}$. Tips: The inverse function is bijective.

c) Condition for the Existence of an Inverse

Most Important Conclusion: A function $f:X \to Y$ has an inverse if and only if $f$ is ==bijective== (one-to-one and onto).