The problem about ideas for studying abstract algebra and … are going to be discussed in the post.
Ideas for abstract algebra
set
A set contains some certain elements. (Not mathematical definition)
relation
Def : (relation)
On a set X , we can define a relation R (not unique). For any two elements in set X, we can tell whether these two elements have a relation, denoted by a ~ b or aRb. We use the former notation more frequently.
equivalence
Def 1: (equivalence relation)
If a relation R on set X (denoted by ~) satisfies three conditions:
Reflexive: a ~ a.
Symmetry: If a ~ b, then b ~ a.
Transitive: If a ~ b, b ~ c, then a ~ c.
then the relation R is called an equivalence relation on set X.
The equivalence relation is an extension of the concept “equality”.
Def 2: (equivalence class)
Consider a set [a] as
then [a] is a equivalence class which contains every element having equivalence relation with a.
Of course, “a” is just a representative of the set [a]. Pick any b from [a], we have [a] = [b].
[a] and [b] have different forms, but is the same in nature.
Any element in X belongs to a unique equivalence class given equivalence relation “ ~ “.
Theorem 1:
An equivalence relation “ ~ “ on a set X gives a division of X.
quotient set
group
subgroup