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Relations and Functions

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Relations and Functions

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Now · 🎯 Learning Objectives

Relations and Functions

🎯 Learning Objectives

Define relations as sets of ordered pairs
Identify properties: reflexive, symmetric, transitive, antisymmetric
Distinguish equivalence relations from partial orders
Classify functions as injective/surjective/bijective
Compose functions and find inverses

6.1 Relations

A relation

R

from

A

B

is a subset of

A \times B

. We write

aRb

for

(a,b) \in R

Properties

Property	Definition	Example on $\mathbb{Z}$
Reflexive	$\forall a: aRa$	$a \leq a$
Symmetric	$aRb \implies bRa$	$a = b$
Transitive	$aRb \land bRc \implies aRc$	$a < b < c$
Antisymmetric	$aRb \land bRa \implies a = b$	$a \leq b$

6.2 Equivalence Relations

A relation that is reflexive, symmetric, and transitive is an equivalence relation. Partitions: An equivalence relation partitions the set into equivalence classes. Example:

a \equiv b \pmod{n}

(congruence mod

n

). Equivalence classes:

\{0\}, \{1\}, \dots, \{n-1\}

modulo

n

6.3 Partial Orders

A relation that is reflexive, antisymmetric, and transitive is a partial order.

Total order: Every pair is comparable ( $\leq$ on $\mathbb{R}$ )
Partial order: Some pairs incomparable ( $\subseteq$ on sets)

Hasse Diagrams

A graphical representation of a partial order.

6.4 Functions

A function

f: A \to B

is a relation where each

a \in A

relates to exactly one

b \in B

Types

Type	Definition	Example
Injective (one-to-one)	$f(a_1) = f(a_2) \implies a_1 = a_2$	$f(x) = 3x$
Surjective (onto)	$\forall b \in B \ \exists a \in A: f(a) = b$	$f(x) = x^3$
Bijective	Both injective and surjective	$f(x) = x+1$

Composition

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

Inverse

f^{-1}(y) = x

iff

f(x) = y

. Exists only if

f

is bijective.

✅ Practice Questions

Q1: Determine if

R = \{(1,1), (1,2), (2,1), (2,2), (3,3)\}

\{1,2,3\}

is an equivalence relation.

Solution

Reflexive? Need $(1,1), (2,2), (3,3)$ — all present. ✓ Symmetric? Check every pair: $(1,2)$ has $(2,1)$ ✓, $(2,1)$ has $(1,2)$ ✓, all self-pairs are symmetric. ✓ Transitive? $(1,2)$ and $(2,1) \to (1,1)$ ✓; $(2,1)$ and $(1,2) \to (2,2)$ ✓. Yes, it's an equivalence relation. Classes: $\{1,2\}$ and $\{3\}$ . Q2: Is $f(x) = x^2$ on $\mathbb{R}$ injective? Surjective? Solution

Not injective: $f(-2) = f(2) = 4$ but $-2 \neq 2$ . Not surjective: $f(x) \geq 0$ , so negative numbers have no preimage. If defined as $f: \mathbb{R}^+ \to \mathbb{R}^+$ , it is bijective. Q3: Find the inverse of $f(x) = 3x + 2$ . Solution

$y = 3x + 2 \implies y - 2 = 3x \implies x = \frac{y-2}{3}$ . So $f^{-1}(y) = \frac{y-2}{3}$ . Join Discord PreviousSet Operation ProofsNextNumber Theory