The 2 important pre-requisites for Abstract Algebra are “abstract thinking”, namely :
- You must not think of “concrete” math objects (geometrical shapes, Integer, Real, complex numbers, polynomials, matrices…), but rather their “generalised” math objects (Group, Ring, Field, Vector Space…).
- Rigourous Proof-oriented rather than computation-oriented.
The foundation of Abstract Algebra is “Set Theory”, make effort to master the basic concepts : eg.
- Equivalence Relation (reflexive, symmetric, transitive),
- Partitioning, Quotient Set, Co-Set
- Necessary condition (“=>”), Sufficient condition (“<=”). Both conditions (“<=>”, aka “if and only if”)
- Proof : “A = B” if and only if ∀a ∈ A, a ∈ B => A ⊂ B, and, ∀b ∈ B, b ∈ A =>B ⊂ A [我中有你, 你中有我 <=>你我合一]
- Check if a function is “well-defined”. (定义良好)
- These concepts / techniques repeat in every branch of Abstract Algebra which deals with all kinds of “Algebraic Structures”, from Group Theory to Ring Theory to Field Theory … to (Advanced PhD Math) Category Theory – aka “The Abstract Nonsense”.