Sets and set operations
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 ∈ A when x is an element of the set A, and x ∉ 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 ∪ B | elements in A or B |
| Intersection | A ∩ B | elements in both A and B |
| Difference | A \ B | elements in A but not in B |
| Complement | A^c | elements outside A in a chosen universal set |
See the operations on a concrete example
Worked example
Track elements through two sets
Let A = {1, 2, 4} and B = {2, 3, 4}.
Then:
A ∪ B = {1, 2, 3, 4}A ∩ B = {2, 4}A \ B = {1}
If the universal set is E = {1, 2, 3, 4, 5}, then A^c = {3, 5}.
Solution
A quick De Morgan check
A note on two extra constructions
The local notes also introduce two useful constructions:
A × B, the Cartesian product, is the set of ordered pairs(a, b).P(A), the power set, is the set of all subsets ofA.
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
A^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, c}` and `B = {b, c, d}`, what is `A ∩ B`?
Try it live
Try it here
Set-operation explorer
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}