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2.2 Functions and relations

Read a function as a special kind of relation, then learn the language of images, preimages, and inverse maps.

Note collections

MATH1090: Set theory

Rigorous course notes on logic, sets, and number construction, written in short linked sections with careful proofs and examples.

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All note collectionsSeries overview

Chapter 1

Logic

Reasoning tools for statements, connectives, and quantifiers.

1.1

1.1 Propositional logic

1.2

1.2 Truth tables and equivalence

1.3

1.3 Quantifiers and negation

Chapter 2

Sets and relations

Basic set language, functions, and relations.

2.1

2.1 Sets and set operations

2.2

2.2 Functions and relations

Chapter 3

Numbers by construction

How natural numbers, integers, and rationals are built, and where Q still falls short.

3.1

3.1 Natural numbers and Peano axioms

3.2

3.2 Induction and recursive arithmetic

3.3

3.3 Integers from equivalence classes

3.4

3.4 Rationals and well-defined operations

3.5

3.5 Gaps in Q and why sqrt(2) is not rational

Chapter 4

Order and completeness

Total order, bounds, supremum and infimum, and the completeness gap between Q and R.

4.1

4.1 Order, bounds, and completeness

Prerequisite: review 2.1 Sets and set operations first, because functions and relations are defined using sets.

What a function is

Definition

A function

A function from XX to YY is a subset of X×YX × Y such that every x in XX is paired with exactly one y in YY.

That is the set-theoretic definition used in the local notes.

Read the basic language

  • The domain is the set of inputs.
  • The target is the set of possible outputs.
  • The graph of the function is the set of pairs (x, y).
  • The image of a set is the set of outputs it reaches.
  • The preimage of a set is the set of inputs that land there.

Common mistake

A function must not send one input to two outputs

A relation may connect one input to many outputs. A function cannot. Every input must have exactly one output.

A first example

Worked example

Read a function from its rule

Take the rule f(x)=x2f(x) = x^2.

The inputs −2-2 and 2 both map to 4, so two different inputs may share the same output.

What is not allowed is one input producing two different outputs.

Solution

Image and preimage

Injective, surjective, bijective

Definition

Three useful words

  • Injective means different inputs never collide.
  • Surjective means every target value is hit.
  • Bijective means both injective and surjective.

The local notes also use this language to describe when an inverse function can exist.

Relations

Definition

A relation

A relation on XX and YY is any subset of X×YX × Y.

If X=YX = Y, we speak of a relation on one set.

Relations may be read with properties such as:

  • reflexive
  • symmetric
  • antisymmetric
  • transitive

Two especially important special cases are:

  • an equivalence relation
  • a partial order

A short example of relation language

Worked example

Equality classes are built from a relation

When a relation is reflexive, symmetric, and transitive, it is an equivalence relation.

Then the equivalence classes group the set into non-overlapping blocks.

Solution

What to remember

Quick check

Quick check

Is x≤yx ≤ y on the real numbers a relation? Is it a function?

Separate the two questions. First ask whether it is a subset of R×RR × R, then ask whether each input is paired with exactly one output.

Solution

Answer

Prerequisites

2.1

2.1 Sets and set operations

Key terms in this unit

More notes in this series

Previous section

2.1 Sets and set operations

Next section

3.1 Natural numbers and Peano axioms