Functions and Groups

The Fundamentals of Functions

A function $f: A \to B$ is a rule assigning each element in a Domain to a unique element in a Codomain.

$$f: A \to B$$

Describes the mapping of elements from the source set to the target set.

$A$=The Domain; set of all possible inputs(set)

$B$=The Codomain; set of all potential outputs(set)

$f(A)$=The Range; the actual set of outputs reached(subset of B)

Domain: The collection of all valid input values (pre-images).

Codomain: The entire set $B$ into which the function is defined to map.

Range: The Image of the entire domain, representing the actual outputs used in $B$.

Classification of Mappings

Mappings are categorized as Injective (one-to-one), Surjective (onto), or Bijective based on their element pairings.

$$\forall a_1, a_2 \in A, f(a_1) = f(a_2) \implies a_1 = a_2$$

The formal definition of an injective (one-to-one) function.

$a_1, a_2$=Distinct elements of the domain(elements)

$f(a)$=The image of element 'a' under mapping 'f'(output)

Injective: Each input maps to a unique, distinct output. Passed by the Horizontal Line Test.

Surjective: Every element in the codomain is mapped to by at least one element in the domain.

Bijective: A one-to-one correspondence where the function is both injective and surjective.

Binary Operations and Group Axioms

A set $G$ with a Binary Operation $\star$ is a group if it satisfies four essential axioms.

$$(G, *) \text{ is a group if } G_1, G_2, G_3, G_4$$

The structural conditions for a set to be an algebraic group.

$G_1$=Closure: result stays in the set(axiom)

$G_2$=Associativity: $(a * b) * c = a * (b * c)$(axiom)

$G_3$=Identity: Existence of element 'e'(axiom)

$G_4$=Inverse: Every element has a partner to reach 'e'(axiom)

Closure: For all $a, b \in G, a \star b \in G$.

Associativity: For all $a, b, c \in G$, $(a \star b) \star c = a \star (b \star c)$.

Identity: There exists $e \in G$ such that $a \star e = e \star a = a$.

Inverse: For every $a \in G$, there is $a^{-1} \in G$ such that $a \star a^{-1} = e$.

Reversal Law: $(a \star b)^{-1} = b^{-1} \star a^{-1}$ — the inverse of a product reverses the order.

Abelian Groups and Group Order

An Abelian group satisfies the Commutative law, meaning the order of operation does not change the result.

$$a * b = b * a$$

Commutativity: The defining feature of Abelian groups. Visualized as symmetry across the main diagonal of a Cayley Table.

Order of a Group: Represented as $|G|$, this is the total count of elements in the set.

Finite vs Infinite: A Finite Group has a specific integer order, while an Infinite Group (like integers under addition) does not.

Composition and Inverse of Functions

The Composition of two functions $f$ and $g$ is written as $(g \circ f)(x) = g(f(x))$ — apply $f$ first, then feed the result into $g$.

$(g \circ f)(x) = g(f(x))$

The output of f becomes the input of g.

$f$=Inner function — applied first

$g$=Outer function — applied to the result of f

Order matters: $g \circ f$ means apply $f$ first, then $g$. Do not confuse with $f \circ g$.

Verification: $(f^{-1} \circ f)(x) = x$ and $(f \circ f^{-1})(x) = x$ — composing a function with its inverse gives the identity.

Shorthand: $f^2(x) = f(f(x))$, $f^3(x) = f(f(f(x)))$, and so on.

An Inverse Function $f^{-1}$ reverses the mapping of $f$: if $f(a) = b$, then $f^{-1}(b) = a$. Only Bijective functions have inverses.

$f^{-1}(f(x)) = x$

Applying f then f⁻¹ (or vice versa) returns the original input.

$f^{-1}$=The inverse function of f

$f$ is not bijective

→

$f^{-1}$ does not exist as a function

$f$ is a linear function $y = mx + c$

→

The inverse is always another linear function

Algebraic method: Set $y = f(x)$, swap $x$ and $y$, solve for $y$ — this gives $f^{-1}(x)$.

Domain-Range swap: Domain of $f^{-1}$ = Range of $f$, and Range of $f^{-1}$ = Domain of $f$.

Graphical symmetry: The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$.

Algebraic Structure Hierarchy

Algebraic structures build up in layers: a Groupoid gains associativity to become a Semi-group, adds an identity to become a Monoid, and adds inverses to become a Group.

Groupoid: A set closed under a binary operation (closure only).

Semi-group: A groupoid where the operation is also associative.

Monoid: A semi-group that has an identity element $e$.

Group: A monoid where every element has an inverse.

Key non-examples: $(\mathbb{N}, +)$ is a semi-group (no identity 0), $(\mathbb{Z}, \times)$ is a monoid (identity 1, but most elements lack integer inverses).