Naive Set Theory
A sets is a collection of mathematical objects. The order doesn’t matter and we ignore duplicates.
We say that A = { 1, 2, 3 } is a set. This means that A is a collection with 1, 2, and 3 as elements. We list all object contained in the set and delimit it with curly braces.
Elements
Some object x is an element of A if x is part of the collection A.
We say x is an element of A or x is in A.
Alternatively, we could say that x is not an element of A or x is not in A.
Operations on Sets
Let A = { 1, 2, 3} and B = { 3, 4, 5 }. We can perform operations of two or more sets, with each operation yielding a new set.
Union
The union operator creates a new set containing all objects that are both in A or in B.
So the union of A and B or A union B will contain the object 1, 2, 3, 4, and 5.
Intersection
The intersection operator creates a new set containing object common to both A and B.
So the intersection of A and B or A intersection B will contain the object 3 since it’s the only object in both sets.
Difference
The difference operator creates a new set with all elements of A but none of the elements of B.
So the difference of A and B or A difference B will contain the object 1, and 2. Note that the object 3 is excluded since it’s also an element of B.
Subsets
We say that A is a subset of B if all elements of A are also elements of B.
Let A = { 1, 2 } and B = { 1, 2, 3, 4, 5 }.
Here A is a subset of B since 1 and 2, all elements in A, are also elements in B.
Empty Set
The empty set, or {}, or ∅ is a set containing no elements. Defining such set might seem confusing, but it’s very useful.
Set-Builder and Set-Roster Notation
All the sets we’ve seen so far are discribed by listing out all its elements, that’s called set-roster notation, set-roaster notation is pretty straight-forward, they’re very easy to read and write. See for example:
The set containing all positive even integers:
A = { 2, 4, 6, 8, 10, … }. The three dots at the end are used to indicate that this pattern continues forever.
The set containing this sequence:
B = { 1, 20, 110, 340, 780, 1500, 2570, …, 500500, 544620 }. The three dots in the middle are used to indacate that the pattern continues up to the last elements listed.
But it doesn’t take long to realize the downsides of this notation, the pattern it presents must be obvious to anyone reading it, otherwise we cannot possibly know what’s the next element going to be. And it’s just tiring to write out all elements of some big set.
For constructing larger and/or more precise sets we can use the set-builder notation. Instead of listing all elements of a particular set, we write a condition that every object must meet in order to be included in the set. See for example:
The set containing all positive even integers:
A = {x | x is an positive even integer } or A = {x is in N | 2x and x > 0 }. Here are two always to write the same set in set-builder notation. We begin by specifying an variable and a condition said variable must meet.
Family of Sets
Sets are a collection of mathematical objects and sets themselves are mathematical objects so we can have sets containing other sets as its elements with no problems.
Let A = { {1, 2}, {3, 4}, {5, 6} }. A only has 3 elements, the set { 1, 2 }, {3, 4}, and { 5, 6 }. The objects 1 or 2 are not elements of A.
Power Sets
The power set of a set contains all possible subsets of that set. Let A = { 1, 2 }, then the power set of A or, as it’s usually denoted, P(A) = { ∅, {1}, {2}, {1, 2} }.
The empty set is a subset of all sets.
Famous Sets
The set of natural numbers contains all counting numbers, 1, 2, 3, and so on, sometimes mathematicians also include 0 in this set.
N = { 1, 2, 3, 4, 5, … }.
The set of integers contains all the positives, negatives, and zero.
Z = { 1, 2, 3, 4, 5, … }.
The set of rational numbers contains all numbers that can be expressed as some number integer a over another integer b.
Q = { …, -2/4, -1, 0, 1, 1/2, 3/2, … }.
The set of irrational numbers, that is, numbers that cannot be expressed as a fraction of two integers don’t have a consistent letter associated with them, most of the time a capital I is used.
I = { …, sqrt(2), 3, pi, … }.
Then there’s the set of real numbers, it’s just the union of the rationals and irrationals.
R = { …, -3, 0, 1, 3/2, sqrt(2), 3, pi, … }.