Maths - Sets

Set

A set is a collection of things, which are called the elements of the set.

Intersection

The intersection of two sets is another set which contains the elements in both the original sets.

Union

The intersection of two sets is another set which contains all the elements from original sets, if an element is in both the original sets then it is only included once, not twice.

Complement

This is everything not in the set. This has to be defined in terms of the universe under discussion.

Cartesian Product

The Cartesian product of two sets is a set of pairs representing every combination of the two sets. This is a very general product that we can define regardless of whether multiplication is a valid operation for the elements of the sets being multiplied.

So this type of product 'generates' a higher dimensional quantity.

So what happens when we multiply two of these two dimensional quantities?

(g1,h1)(g2,h2) = ?

Perhaps it might generate a 4 dimensional quantity:

(g1,h1)(g2,h2) = (g1,h1,g2,h2)

Or we might be able to define a multiplication which would make this equivalent to complex number multiplication:

(g1,h1)(g2,h2) = (g1g2 - h1h2 , g1h2 + g2h1)

Or we may require a different definition,as an example, lets take the dihedral group D3, that we looked at on this page. We might represent any transform as a pair like:

(ma,ra)

Consisting of one reflection and one rotation, we can represent any element of this group by such a pair.

Lets take another example, imagine that we have a group G and a subgroup H with elements g1,g2… and h1,h2…

In this case we can now define multiplication as g1 H * g2 H = g1 g2 H

g1 H	g2 H	g1 g2 H
left coset g1 of H	left coset g2 of H	left coset g1,g2 of H

Product

If multiplying individual elements of the first set by elements of the second set is valid, then the product no longer needs to be represented by pairs, in this case the product contains the product of every combination of the two sets. If we are combining two groups then we may be able to use an external or internal product as described on this page.

Equivalence Relation

Generalisation of equality to set theory.

If two sets are exatly equal, this may not be all that interesting from a mathematical point of view. It is often more interesting when two sets are not identical but preserve some form of structure when mapping between them.

	type	category notation
tighter	Equality
	Isomorphism	1 = GF FG = 1
	Equilivance	1GF FG1
looser	Adjunctions	1 => GF FG => 1

Equivalence:

Set Notation	Alternative Notation	Property
(a,a)R for all aS	aa	Reflective
(a,b)R implies (b,a)R	ab implies ba	Symmetric
(a,b)R and (b,c)R implies (a,c)R	ab and bc implies ac	Transitive

Sequence in a set

A function from N (the set of positive integers) to the set.

In other words each element of the set is assigned a number.

Notation:

{a_n}_n=1^∞

Indexing sets

An index set is a set used to label the elements of another set using a surjective function.

An alternative approach is to attach a name to each element of a collection of sets.

Notation:

{a_α}_α∈A

Related Topics

Open and closed sets.

Set Symbols

1:1 Correspondence Definition

Function Definition

Morphism (homomorphism) Definition

Representation Definition