Modern algebra chapter 1

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


CHAPTER 1

α is a mapping from S to T

 

 

     α

S -> T = α : S -> T

 

 

 

α(x) is called the image if x under the mapping α (x is element of S)

 

α(S) is called the image of α

 

 

 

 

above figure is not mapping because x assigns two elements and y assigns no element

 

 

ι(iota)

 

ι(x) = x ( x ∈ S )

 

ιs：set S is being considered

 

 

 

    α

x -> y or x ↦ y

 

 

 

α : S -> T and A is subset of S, then α(A) is a subset of T

 

α(A) = {α(x) : x ∈ A}

 

 

 

 

 

<onto=surjection>

 

 

Above left figure, α is said to be ‘onto’ but right figure α is not ‘onto’

That means all elements of T have at least one x ∈ S, α(x) = y

 

<one-to-one=injection>

if x1 ≠ x2, then α(x1) ≠ α(x2) (x1, x2, ∈ S)

 

 

left diagram is one-to-one but right is not one-to-one

<both one-to-one and onto = bijection>