Understanding Relations and Functions in Mathematics

Relations and functions form a foundational part of mathematical study, linking the concepts of sets to the structure and behaviour of mappings between sets. In this topic, the analysis focuses on the formal mechanisms of associating elements across sets and systematically examines the structural properties of these associations.

Definition of Relations Between Two Sets

Given two non-empty sets $A$ and $B$, the Cartesian product $A \times B$ denotes the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. A relation $R$ from $A$ to $B$ is any subset of $A \times B$, that is, $R \subseteq A \times B$. If $(a, b) \in R$, then $a$ is said to be related to $b$ under $R$.

If $B = A$, then $R$ is called a relation on $A$. In terms of notation, $(a, b) \in R$ is denoted by $a\,R\,b$.

Classification of Relations: Reflexivity, Symmetry, and Transitivity

A relation $R$ on a set $A$ possesses specific properties, outlined formally as follows:

Reflexive Relation: $R$ is reflexive if for every $a \in A$, $(a, a) \in R$.

Symmetric Relation: $R$ is symmetric if for every $a, b \in A$, $(a, b) \in R \implies (b, a) \in R$.

Transitive Relation: $R$ is transitive if for every $a, b, c \in A$, $(a, b) \in R$ and $(b, c) \in R \implies (a, c) \in R$.

Equivalence Relations and Equivalence Classes

A relation $R$ on $A$ that is reflexive, symmetric, and transitive is termed an equivalence relation. For such a relation, the set $[a] = \{ x \in A : (a, x) \in R \}$ is called the equivalence class of $a$.

The set of all distinct equivalence classes forms a partition of $A$, such that every element of $A$ belongs to exactly one equivalence class.

Definition of Functions (Mappings) As Special Cases of Relations

A function $f$ from $A$ to $B$, denoted as $f: A \to B$, is a relation such that for every $a \in A$, there exists a unique $b \in B$ with $(a, b) \in f$. The set $A$ is called the domain, $B$ is the codomain, and the set $\{b \in B \mid (a, b) \in f \text{ for some } a \in A\}$ is termed the range.

Theorems on the Algebra of Functions

Let $f: A \to B$ and $g: B \to C$ be functions. The composition $g \circ f: A \to C$ is defined by $(g \circ f)(a) = g(f(a))$, for all $a \in A$.

Theorem: The composition of functions is associative.
Let $f: A \to B$, $g: B \to C$, $h: C \to D$ be functions. Then, for all $a \in A$, \[ (h \circ (g \circ f))(a) = ((h \circ g) \circ f)(a) \] Step 1: Compute $g \circ f$.
$(g \circ f)(a) = g(f(a))$.
Step 2: Substitute into $h$.
$(h \circ (g \circ f))(a) = h((g \circ f)(a)) = h(g(f(a)))$.
Step 3: Now compute $h \circ g$.
$(h \circ g)(b) = h(g(b))$ for all $b \in B$.
Step 4: $\big((h \circ g) \circ f\big)(a) = (h \circ g)(f(a)) = h(g(f(a)))$.
Step 5: Therefore, $(h \circ (g \circ f))(a) = ((h \circ g) \circ f)(a)$.
Result: Composition of functions is associative.

Injective, Surjective, and Bijective Functions

Let $f: A \to B$ be a function. $f$ is injective (one-to-one) if for all $a_1, a_2 \in A$, $f(a_1) = f(a_2) \implies a_1 = a_2$.

$f$ is surjective (onto) if for every $b \in B$, there exists $a \in A$ such that $f(a) = b$.

$f$ is bijective if it is both injective and surjective. In this case, the inverse function $f^{-1}: B \to A$ exists and is unique.

Worked Examples on Relations and Functions

Given: Let $A = \{1,2,3\}$. Define the relation $R$ on $A$ by $R = \{(a, b) \mid a - b \text{ is even}\}$.
Substitution: For $a, b \in \{1,2,3\}$, compute $a - b$: $(1,1)$, $(1,3)$, $(2,2)$, $(3,1)$, $(3,3)$ have $a-b$ even.
Simplification: $R = \{ (1,1), (1,3), (2,2), (3,1), (3,3) \}$.
Final result: $R$ is reflexive and symmetric but not transitive.

Given: Define $f: \mathbb{R} \to \mathbb{R}$, $f(x) = 2x + 5$.
Substitution: For $x_1, x_2 \in \mathbb{R}$, $f(x_1) = f(x_2) \implies 2 x_1 + 5 = 2 x_2 + 5$.
Simplification: $2x_1 = 2x_2 \implies x_1 = x_2$.
Final result: $f$ is injective. For surjectivity, let $y \in \mathbb{R}$, $y = 2x + 5 \implies x = \frac{y-5}{2} \in \mathbb{R}$. Thus, $f$ is bijective.

Graphical Representation of Relations and Functions

A function $f: \mathbb{R} \to \mathbb{R}$ can be graphically represented as the set of points $(x, f(x))$ in the plane. A relation can have multiple $y$-values for a given $x$, but a function by definition associates a unique $y$ to each $x$.

Worked Example on Graphs of Functions

Given: Consider $f(x) = x^2$ for $x \in \mathbb{R}$.
Substitution: Select $x = -2, 0, 2$; compute $f(-2) = 4$, $f(0) = 0$, $f(2) = 4$.
Final result: Points $(-2,4),\ (0,0),\ (2,4)$ lie on the graph of $f$, and the set $\{ (x, x^2): x \in \mathbb{R} \}$ defines the graphical representation.

For systematic practice and further conceptual exploration, refer to All About Relations And Functions.