mapping/function
A mapping or function α : A → B is a rule that assigns to every element a of A exactly one element α(a) of B
- well defined or function property of a rule.
domain
A is called the domain of the mapping α written D(α).
image
(element)
α(a) is called the image of element a.
image
(mapping)
α(A) = {α(a)|a ∈ A} = Im(α) is called the range or image of mapping α.
Im(α) is a subset of B.
graph
Graph of G is G(α) = {(a, b)|a ∈ A, b ∈ B and b ∈ α(a)} = {(a, α(a))|a ∈ A}.
binary operation
If A = S × S and B = S then the mapping is a binary operation on the set S.
well defined mapping
(create)
(i) Specifying the domain A and the co-domain B of α.
(ii) Specifying exactly one element α(a) for each a in A.
equal mapping
α : A → B and β : A → B are equal if and only if α(a) = β(a) for each a ∈ A.
identity mapping
α : A → B is called the identity mapping if α(a) = a for each a in A and is denoted by 1A.
Obviously A ⊆ B in this case.
injective mapping
α : A → B is said to be injective or one-to one if α(a) = α(b) implies that a = b.
(the horizontal line test)
surjective mapping
α : A → B is onto or surjective if for each b ∈ B we can find a ∈ A such that α(a) = b.
That is B = Im(α).
composition mapping
Given α : A → B and β : B → C, then the composition mapping βα : A → C is defined by βα(a) = β(α(a)) for all a ∈ A.
αβ DNE in this case.
Note: associative but not commutative.
inverse mapping
If α : A → B is a mapping , a mapping β : B → A is called inverse of α if βα = 1A and αβ = 1B. We say the mapping α is invertible.
Note β(b) = a iff α(a) = b.
If β exists it is unique and we can write it as α^(−1).
numerically equivalent mapping
If α : A → B is a bijection and A and B are both finite, then |A|=|B|, hence we say A and B are numerically equivalent
one-to-one
correspondence
If there exists a bijection from set A onto set B , we say A and B are in one-to-one correspondence
Invertibility Theorem
A mapping α : A → B is invertible if and only if it is a bijection.
Thus if α : A → B is a bijection then α−1 exists as a mapping from B to A and this inverse is unique. Further if a mapping is invertible then it is a bijection.
