1: Sets and functions

Section 1: Mathematical Preliminaries: Sets and Functions

Sets and functions form the foundational language of abstract algebra. A set is a well-defined collection of distinct objects, called elements or members. Sets are typically denoted by capital letters (e.g., $A, B$) and elements by lowercase letters (e.g., $a, b$). We write $a \in A$ to indicate "a is an element of A". Common sets include the natural numbers $\mathbb{N}$, integers $\mathbb{Z}$, rational numbers $\mathbb{Q}$, real numbers $\mathbb{R}$, and complex numbers $\mathbb{C}$.

Key Set Operations:

• Union ($A \cup B$): Elements in $A$, $B$, or both. $A \cup B = \{ x \mid x \in A \text{ or } x \in B \}$.
• Intersection ($A \cap B$): Elements common to both $A$ and $B$. $A \cap B = \{ x \mid x \in A \text{ and } x \in B \}$.
• Set Difference ($A \setminus B$ or $A - B$): Elements in $A$ but not in $B$. $A \setminus B = \{ x \mid x \in A \text{ and } x \notin B \}$.
• Complement ($A^c$): Elements not in $A$ (relative to a universal set $U$). $A^c = U \setminus A$.
• Cartesian Product ($A \times B$): Set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

A function (or mapping) $f$ from a set $A$ (the domain) to a set $B$ (the codomain), denoted $f: A \to B$, assigns to each element $a \in A$ exactly one element $b \in B$, written $b = f(a)$. The element $b$ is the image of $a$. The set of all images $\{ f(a) \mid a \in A \} \subseteq B$ is the range (or image) of $f$.

Function Properties:

• Injective (One-to-One): Distinct inputs produce distinct outputs. Formally: if $f(a_1) = f(a_2)$ then $a_1 = a_2$ for all $a_1, a_2 \in A$.
• Surjective (Onto): Every element in the codomain is an image of some element in the domain. Formally: for every $b \in B$, there exists $a \in A$ such that $f(a) = b$ (i.e., the range equals the codomain).
• Bijective: A function is bijective if it is both injective and surjective. Bijective functions have inverse functions $f^{-1}: B \to A$, defined by $f^{-1}(b) = a$ if and only if $f(a) = b$.

Given functions $f: A \to B$ and $g: B \to C$, their composition is the function $g \circ f: A \to C$ defined by $(g \circ f)(a) = g(f(a))$ for all $a \in A$. Composition is associative: $h \circ (g \circ f) = (h \circ g) \circ f$. For a bijective function $f$, the inverse satisfies $f^{-1} \circ f = \text{id}_A$ and $f \circ f^{-1} = \text{id}_B$, where $\text{id}$ denotes the identity function.