HNDIT: Introduction to Sets

Introduction to Sets

What is a set? Well, simply put, it's a collection.

First you specify a common property among "things" (this word will be defined later) and then you gather up all the "things" that have this common property.

Notation

There is a fairly simple notation for sets. You simply list each element, separated by a comma, and then put some curly brackets around the whole thing.

The curly brackets { } are sometimes called "set brackets" or "braces".

Numerical Sets

So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers?

Set of even numbers: {..., -4, -2, 0, 2, 4, ...}
Set of odd numbers: {..., -3, -1, 1, 3, ...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Positive multiples of 3 that are less than 10: {3, 6, 9}

Universal Set

At the start we used the word "things" in quotes. We call this the universal set. It's a set that contains everything. Well, not exactly everything. Everything that is relevant to the problem you have.

Equality

Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, you may have to examine them closely!

Example: Are A and B equal where:

A is the set whose members are the first four positive whole numbers
B = {4, 2, 1, 3}

Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so: Yes, they are!

And the equals sign (=) is used to show equality, so you would write:

A = B

Subsets

When we define a set, if we take pieces of that set, we can form what is called a subset.
So for example, we have the set {1, 2, 3, 4, 5}. A subset of this is {1, 2, 3}. Another subset is {3, 4} or even another, {1}. However, {1, 6} is not a subset, since it contains an element (6) which is not in the parent set. In general:

A is a subset of B if and only if every element of A is in B.

Proper Subsets

If we look at the definition of subsets and let our mind wander a bit, we come to a weird conclusion.
Let A be a set. Is every element in A an element in A? (Yes, I wrote that correctly.)

Well, yes of course, right?

So wouldn't that mean that A is a subset of A?
This doesn't seem very proper, does it? We want our subsets to be proper. So we introduce (what else but) proper subsets.
A is a proper subset of B if and only if every element in A is also in B, and there exists at least one element in B that is not in A.
This little piece at the end is only there to make sure that A is not a proper subset of itself. Otherwise, a proper subset is exactly the same as a normal subset.

Example:

{1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}.

Example:

{1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set.

You should notice that if A is a proper subset of B, then it is also a subset of B.

Empty (or Null) Set

It is a set with no elements.This is known as the Empty Set (or Null Set). There aren't any elements in it. Not one. Zero. It is represented by

Or by {} (a set with no elements) Some other examples of the empty set are the set of countries south of the south pole. So what's so weird about the empty set?

Order

No, not the order of the elements. In sets it does not matter what order the elements are in.

Example: {1,2,3,4) is the same set as {3,1,4,2}

When we say "order" in sets we mean the size of the set.

Just as there are finite and infinite sets, each has finite and infinite order.
For finite sets, we represent the order by a number, the number of elements.

Example, {10, 20, 30, 40} has an order of 4.

Table of set theory symbols

Symbol	Symbol Name	Meaning / definition	Example
{ }	set	a collection of elements	A={3,7,9,14}, B={9,14,28}
A ∩ B	intersection	objects that belong to set A and set B	A ∩ B = {9,14}
A ∪ B	union	objects that belong to set A or set B	A ∪ B = {3,7,9,14,28}
A ⊆ B	subset	subset has fewer elements or equal to the set	{9,14,28} ⊆ {9,14,28}
A ⊂ B	proper subset / strict subset	subset has fewer elements than the set	{9,14} ⊂ {9,14,28}
A ⊄ B	not subset	left set not a subset of right set	{9,66} ⊄ {9,14,28}
A ⊇ B	superset	set A has more elements or equal to the set B	{9,14,28} ⊇ {9,14,28}
A ⊃ B	proper superset / strict superset	set A has more elements than set B	{9,14,28} ⊃ {9,14}
A ⊅ B	not superset	set A is not a superset of set B	{9,14,28} ⊅ {9,66}
2^A	power set	all subsets of A
Ƥ (A)	power set	all subsets of A
A = B	equality	both sets have the same members	A={3,9,14}, B={3,9,14}, A=B
A^c	complement	all the objects that do not belong to set A
A \ B	relative complement	objects that belong to A and not to B	A={3,9,14}, B={1,2,3}, A-B={9,14}
A - B	relative complement	objects that belong to A and not to B	A={3,9,14}, B={1,2,3}, A-B={9,14}
A ∆ B	symmetric difference	objects that belong to A or B but not to their intersection	A={3,9,14}, B={1,2,3}, A ∆ B={1,2,9,14}
A ⊖ B	symmetric difference	objects that belong to A or B but not to their intersection	A={3,9,14}, B={1,2,3}, A ⊖ B={1,2,9,14}
a∈A	element of	set membership	A={3,9,14}, 3 ∈ A
x∉A	not element of	no set membership	A={3,9,14}, 1 ∉ A
(a,b)	ordered pair	collection of 2 elements
A×B	Cartesian product	set of all ordered pairs from A and B
\|A\|	cardinality	the number of elements of set A	A={3,9,14}, \|A\|=3
#A	cardinality	the number of elements of set A	A={3,9,14}, #A=3
	aleph-null	infinite cardinality of natural numbers set
	aleph-one	cardinality of countable ordinal numbers set
Ø	empty set	Ø = { }	C = {Ø}
U	universal set	set of all possible values
ℕ₀	natural numbers / whole numbers set (with zero)	ℕ₀ = {0,1,2,3,4,...}	0 ∈ ℕ₀
ℕ₁	natural numbers / whole numbers set (without zero)	ℕ₁ = {1,2,3,4,5,...}	6 ∈ ℕ₁
ℤ	integer numbers set	ℤ = {...-3,-2,-1,0,1,2,3,...}	-6 ∈ ℤ
ℚ	rational numbers set	ℚ = {x \| x=a/b, a,b∈ℕ}	2/6 ∈ ℚ
ℝ	real numbers set	ℝ = {x \| -∞ < x <∞}	6.343434 ∈ ℝ
ℂ	complex numbers set	ℂ = {z \| z=a+bi, -∞<a<∞, -∞<b<∞}	6+2i ∈ ℂ