Notes on Abstract Algebra

1 Functions Definition 1.1. Given sets S and T , suppose there is a subset G ⊆ S × T with the following properties: • If (s1, t1) ∈ G and (s2, t2) ∈ G and s1 = s2 then t1 = t2; • For each s ∈ S, there is an element (s, t) ∈ G. Then for each s ∈ S, there is exactly one element α(s) ∈ T so that (s, α(s)) ∈ G. This defines a function α, with domain S and target T , which can be denoted α : S → T . Theorem 1.2. Given S = Ø, and a function α : S → T , the following are equivalent: 1. For all s1, s2 ∈ S, if s1 = s2, then α(s1) = α(s2) (α has the one-to-one property); 2. For any set C and any functions γ : C → S, δ : C → S, if α◦γ : C → T and α◦δ : C → T are the same function, then γ = δ (α has the left cancellable property); 3. There is a function β : T → S so that β ◦ α : S → S is equal to the identity function ι : S → S (α has a left inverse).