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- A set function is a function whose domain is a collection of sets. In many instances in real analysis, a set function is a function which associates an affinely extended real number to each set in a collection of sets.
mathworld.wolfram.com/SetFunction.html
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Jul 21, 2022 · If $f$ has domain $D$, we can form the set $S = \{f(x) \mid x \in D \}$. This, too, has two names. Some folks call it the "range", and others call it the image. Because of the name-clash with the other use of "range", it's probably a good idea to stick with "codomain" and "image" and leave "range" out of it.
- elementary set theory
When we say that f f is a function from X X into Y Y then we...
- elementary set theory
Apr 17, 2022 · (a) Let \(S = \{1, 2, 3, 4\}\). Define \(F: S \to \mathbb{N}\) by \(F(x) = x^2\) for each \(x \in s\). What is the range of the function \(F\) and what is \(F(S)\)? How do these two sets compare? Now let \(A\) and \(B\) be sets and let \(f: A \to B\) be an arbitrary function from \(A\) to \(B\). (b) Explain why \(f(A) = \text{range}(f)\).
- Input, Relationship, Output
- Some Examples of Functions
- Names
- The "X" Is Just A Place-Holder!
- Sometimes There Is No Function Name
- Relating
- What Types of Things Do Functions Process?
- A Function Is Special
- The Two Important Things!
- Vertical Line Test
We will see many ways to think about functions, but there are always three main parts: 1. The input 2. The relationship 3. The output
But we are not going to look at specific functions ... ... instead we will look at the general ideaof a function.
First, it is useful to give a function a name. The most common name is "f", but we can have other names like "g" ... or even "marmalade" if we want. But let's use "f": We say "f of x equals x squared" what goes intothe function is put inside parentheses () after the name of the function: So f(x) shows us the function is called "f", and "x" goes in ...
Don't get too concerned about "x", it is just there to show us where the input goes and what happens to it. It could be anything!
Sometimes a function has no name, and we see something like: y = x2 But there is still: 1. an input (x) 2. a relationship (squaring) 3. and an output (y)
At the top we said that a function was likea machine. But a function doesn't really have belts or cogs or any moving parts - and it doesn't actually destroy what we put into it! A function relatesan input to an output. Saying "f(4) = 16" is like saying 4 is somehow related to 16. Or 4 → 16
So we need something more powerful, and that is where setscome in: Each individual thing in the set (such as "4" or "hat") is called a member, or element. So, a function takes elements of a set, and gives back elements of a set.
But a function has special rules: 1. It must work for everypossible input value 2. And it has only one relationshipfor each input value This can be said in one definition:
When a relationship does not follow those two rules then it is not a function ... it is still a relationship, just not a function.
On a graph, the idea of single valuedmeans that no vertical line ever crosses more than one value. If it crosses more than once it is still a valid curve, but is not a function. Some types of functions have stricter rules, to find out more you can read Injective, Surjective and Bijective
1 Sets. The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be ex-pressible in terms of functions. We will not de ne what a set is, but take as a basic (unde ned) term the idea of a set X and of membership x 2 X (x is an element of X).
Sets. A set is a collection of objects, called the elements or members of the set. The objects could be anything (planets, squirrels, characters in Shakespeare's plays, or other sets) but for us they will be mathematical objects such as numbers, or sets of numbers.
Mar 24, 2021 · Definition: Function. Let A A and B B be nonempty sets. A function from A A to B B is a rule that assigns to every element of A A a unique element in B B. We call A A the domain, and B B the codomain, of the function. If the function is called f f, we write f: A → B f: A → B.
Sep 1, 2013 · When we say that f f is a function from X X into Y Y then we mean to say that f f is a set of ordered pairs (x, y) ( x, y) such that x ∈ X x ∈ X and y ∈ Y y ∈ Y, and the following holds: For every x ∈ X x ∈ X there is some y ∈ Y y ∈ Y such that (x, y) ∈ f ( x, y) ∈ f. If (x, y) ∈ f ( x, y) ∈ f and (x,y′) ∈ f ( x, y ′) ∈ f then y = y′ y = y ′.
