Relations & Functions

Relations and functions describe how inputs are paired with outputs. Every function is a relation, but not every relation is a function, because a function gives each input exactly one output.

In CSEC, this topic often asks you to move between representations: ordered pairs, mapping diagrams, tables, equations, and graphs. Keep track of the domain, codomain, and range, because those words describe the input choices, possible targets, and actual outputs.

A Relation Is a Connection¶

A relation is simply a connection between two sets of things. It pairs up elements from one set with elements from another set.

Real-world examples:

• Each student paired with their test score
• Each city paired with its temperature
• Each book paired with its author
• Each date paired with the day's closing stock price

Sets and Notation¶

Before we formalize relations, understand the vocabulary:

• Set: A collection of objects (written in curly braces)

  • Example: $A = \{1, 2, 3, 4, 5\}$ (the set of single digits 1-5)
  • Example: $B = \{a, e, i, o, u\}$ (vowels)

• Ordered pair: Two elements in a specific order, written as $(x, y)$

  • $(3, 6)$ is different from $(6, 3)$
  • First element: input
  • Second element: output

• Cartesian product: All possible ordered pairs from two sets

  • If $A = \{1, 2\}$ and $B = \{a, b\}$
  • Then $A \times B = \{(1,a), (1,b), (2,a), (2,b)\}$ (4 pairs total)

Defining a Relation Formally¶

A relation from set $A$ to set $B$ is a subset of the Cartesian product $A \times B$.

In simpler terms: Pick some (but not necessarily all) ordered pairs from $A \times B$.

Key Properties of Relations¶

Domain — The Input Set¶

The domain is the set of all first elements (inputs).

Codomain — The Target Set¶

The codomain is the set we're pairing TO (the "target" set we might use, whether we actually use all elements or not).

Range (or Image) — The Output Set¶

The range is the set of all second elements (outputs) that actually appear.

Types of Relations¶

One-to-One (Injective)¶

Each input connects to exactly one output, and no two inputs connect to the same output.

Example: Student → Student ID (each student has one unique ID)

Many-to-One¶

Multiple inputs can connect to the same output.

Example: City → Country (many cities in one country)

One-to-Many¶

One input connects to multiple outputs.

Example: Person → Hobbies (one person has many hobbies)

Many-to-Many¶

Inputs and outputs can have multiple connections.

Example: Students → Subjects (many students in many subjects)

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Part 2: Functions — Special Relations¶

What Makes a Function Special?¶

A function is a special type of relation with ONE strict rule:

Each input must have EXACTLY ONE output.

This means:

• ✓ Input 3 connects to output 9 (good)
• ✓ Input 5 connects to output 25 (good)
• ✗ Input 3 connects to outputs 9 AND 10 (bad! not a function)
• ✗ Input 5 connects to outputs 25 AND 5 (bad! not a function)

However: It's okay if two different inputs map to the same output (many-to-one is fine for functions).

Function Notation¶

When we have a function, we use special notation to show the rule.

Standard Notation¶

$f: A \to B$

Read as: "Function $f$ from set $A$ to set $B$"

• $A$ = domain (the inputs)
• $B$ = codomain (the target outputs)
• The rule defines which output goes with each input

Function Rule Notation¶

$f(x) = \text{formula}$

This means:

• $f$ is the name of the function
• $x$ is the input variable
• $f(x)$ is the output (what you get after applying the rule)

Examples:

• $f(x) = 2x + 1$ means: "Take the input $x$, double it, and add 1"
• $g(x) = x^2$ means: "Take the input $x$ and square it"
• $h(t) = 5t$ means: "Take the input $t$ and multiply by 5"

Evaluating Functions — Finding Outputs¶

To evaluate a function at a specific value, substitute that value in for $x$.

Functions vs. Relations — The Key Difference¶

The vertical line test distinguishes functions from non-functions:

Vertical line test rule: If you draw a vertical line at any $x$-value on the graph, it crosses the graph at most ONCE, the relation is a function.

• Crosses once: function ✓
• Crosses twice or more: not a function ✗
• Doesn't cross: that $x$-value isn't in the domain