2.3 Functions Flashcards
When are two functions equal
When they have the same domain and the same codomain and map each element of their common domain to the same element in their common codomain. If either the domain of the codomain changes, we obtain a different function
Definition of one-to-one functions
A function is said to be one-to-one, or an injection, if and only if f(a)= f(b) implies that a = b for all a and b in the domain of f.
A function is on-to-one if and only if f(a) ≠ f(b) whenever a ≠ b
Expressing one-to-one using quantifiers
Definition of onto
A function from A to B is called onto, or a surjection, if and only if for every element in b ∈ B there is an eelement a ∈ A with f(a) = b
Every element in the codomain has a preimage in the domain
Expressing onto using quantifiers
Definition of bijection
A function is a one-to-one correspondence, or a bijection if it is both one-to-one and onto
How to show that f is injective
Show that if f (x) = f (y) for arbitrary x, y ∈ A with x ≠ y, then x = y
How to show that f is not injective
Find particular elements x, y ∈ A such that x ≠ y and f (x) = f (y)
How to show that f is surjective
Consider an arbitrary element y ∈ B and find an element x ∈ A such that f (x) = y
How to show that f is not surjective
Find a particular y ∈ B such that f (x) ≠ y for all x ∈ A
The inverse of f(a) = b
The composition of the functions f and g
(f ◦ g)(a) = f (g(a))
The graph of a function from set A to B (set notation)
The set of ordered pairs {(a, b) : a ∈ A and f (a) = b}
What type of function is this
One-to-one, not onto
What type of function is this
Onto, not one-to-one
What type of function is this
one-to-one and onto
What type of function is this
Neither one-to-one nor onto
What type of function is this
Not a function
