Answer:
The word "image" is used in three related ways. In these definitions, f : X → Y is a function from the set X to the set Y.
Image of an element Edit
If x is a member of X, then the image of x under f, denoted f(x),[1] is the value of f when applied to x. f(x) is alternatively known as the output of f for argument x.
Image of a subset Edit
The image of a subset A ⊆ X under f, denoted {\displaystyle f[A]}{\displaystyle f[A]}, is the subset of Y which can be defined using set-builder notation as follows:[2]
{\displaystyle f[A]=\{f(x)\mid x\in A\}}{\displaystyle f[A]=\{f(x)\mid x\in A\}}
When there is no risk of confusion, {\displaystyle f[A]}{\displaystyle f[A]} is simply written as {\displaystyle f(A)}f(A). This convention is a common one; the intended meaning must be inferred from the context. This makes f[.] a function whose domain is the power set of X (the set of all subsets of X), and whose codomain is the power set of Y. See § Notation below for more.
Image of a function Edit
The image of a function is the image of its entire domain, also known as the range of the function.[3]
Generalization to binary relations Edit
If R is an arbitrary binary relation on X×Y, then the set { y∈Y | xRy for some x∈X } is called the image, or the range, of R. Dually, the set { x∈X | xRy for some y∈Y } is called the domain of R.
Answer:
Set Theory
It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other. We can use these sets understand relationships between groups, and to analyze survey data.
Basics
An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a set.
SET
A set is a collection of distinct objects, called elements of the set
A set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets.
EXAMPLE 1
Some examples of sets defined by describing the contents:
The set of all even numbers
The set of all books written about travel to Chile
Answers
Some examples of sets defined by listing the elements of the set:
{1, 3, 9, 12}
{red, orange, yellow, green, blue, indigo, purple}
A set simply specifies the contents; order is not important. The set represented by {1, 2, 3} is equivalent to the set {3, 1, 2}.
NOTATION
Commonly, we will use a variable to represent a set, to make it easier to refer to that set later.
The symbol ∈ means “is an element of”.
A set that contains no elements, { }, is called the empty set and is notated ∅
EXAMPLE 2
Let A = {1, 2, 3, 4}
To notate that 2 is element of the set, we’d write 2 ∈ A
Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris’s collection is a set, we can also say it is a subset of the larger set of all Madonna albums.
