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2.1 Sets and set operations

Understand membership, subsets, and the main set operations through concrete examples.

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: if you want the notation in this unit to feel familiar, review 1.1 Propositional logic first. Set membership is often read with logical language in the background.

What a set is

Definition

A set

A set is a collection of things.

We write x∈Ax ∈ A when x is an element of the set AA, and x∉Ax ∉ A when it is not.

Two sets are equal when they have exactly the same elements.

The main operations

| Operation | Symbol | Read as | | --- | --- | --- | | Union | A∪BA ∪ B | elements in AA or BB | | Intersection | A∩BA ∩ B | elements in both AA and BB | | Difference | A \ B | elements in AA but not in BB | | Complement | AcA^c | elements outside AA in a chosen universal set |

See the operations on a concrete example

Worked example

Track elements through two sets

Let A=1,2,4A = {1, 2, 4} and B=2,3,4B = {2, 3, 4}.

Then:

  • A∪B=1,2,3,4A ∪ B = {1, 2, 3, 4}
  • A∩B=2,4A ∩ B = {2, 4}
  • A B=1A \ B = {1}

If the universal set is E=1,2,3,4,5E = {1, 2, 3, 4, 5}, then Ac=3,5A^c = {3, 5}.

Solution

A quick De Morgan check

A note on two extra constructions

The local notes also introduce two useful constructions:

  • A×BA × B, the Cartesian product, is the set of ordered pairs (a, b).
  • P(A), the power set, is the set of all subsets of AA.

These are worth recognizing early, even if you only use them in a few places in this unit.

Common mistake

Common mistake

Do not confuse complement and difference

AcA^c depends on a universal set. A \ B depends on a second set. They are not the same idea.

Quick check

Quick check

If A=a,b,cA = {a, b, c} and B=b,c,dB = {b, c, d}, what is A∩BA ∩ B?

Work directly from the definition of intersection: keep only the elements that belong to both sets.

Solution

Answer

Pause and test the idea

Read and try

Compare one pair of sets

The live explorer lets you move elements in and out of A and B and watch the resulting operations update immediately.

Set A

Set B

Union

{1, 2, 3, 4}

Intersection

{2, 4}

Difference A \ B

{1}

Prerequisites

This section can be read on its own.

Key terms in this unit

More notes in this series

Previous section

1.3 Quantifiers and negation

Next section

2.2 Functions and relations