Modern algebra chapter 2

reference: Modern Algebra An Introduction (John R. Durbin)

CHAPTER 2

 

<composition>

α: S -> T and β: T -> U

equal

(α∘β)(x) = β(α(x)) (for each x ∈ S)

 

(α∘β)(x) ≠ (β∘α)(x)

 

(a) if α and β are onto, then β∘α is onto.

(b) if β∘α is onto, then β is onto.

(c) if α and β are one-to-one, then β∘α is one-to-one

(d) if β∘α is one-to-one, then α is one-to-one.

 

<inverse>

β: T -> S in an inverse of α: S -> T

 

so, β∘α = ιS

 

if a mapping is invertible, then its inverse is unique