Functions and Sets: Get Help Understanding Them

In summary, understanding functions using sets can be helpful in fully grasping their meaning. Sets can be used as a foundation for defining other mathematical objects, such as functions, and understanding this relationship can provide a deeper understanding of mathematics. Sets are the building blocks of mathematics, and by using them, we can define and understand various mathematical concepts.
  • #1
cam875
228
0
Ive never been taught functions using sets and I have been told that to really understand the real meaning of them it is helpful to work them out and understand them with sets and stuff. I am not sure if I am on the right track or confused but I am sure someone here can help me out. I have a basic understanding of sets and elements and all that so i should be able to follow along.
thanks in advance.
 
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  • #2
I am not sure how often the set definition of functions is helpful. There is also a notion of a function inducing a function from the power set of the domain into the power set of the range. It is helpful to first give the definition of a relation.

Definition: Catesian Product
Let A and B be sets. AxB is called the Cartesian product of A and B.
AxB contains all ordered pairs of the form (a,b) were a and be are respectively elements of A and B.

Definition: relation from A to B (set version)
Let A and B be sets.
R is a relation from A to B if R is a subset of AxB

Definition: Domain (of a relation)
Let R be a relation from A to B
The domain of R, Dom(R) is the set of all elements a in A such that there exists a b in B so that
(a,b) is in R.

Definition: range (of a relation)
Let R be a relation from A to B
The range of R, Rng(R) is the set of all elements b in B such that there exists a in A so that
(a,b) is in R.

Definition: function from A to B (set version)
let A and B be sets. A function f from A to B is a relation from A to B such that for each a in A there exist exactly one b in B such that (a,b) is in f.

Often is is helpful to break this condition in two parts.
Dom(f)=A (for each a in A there exist at least one b in B such that (a,b) is in f)
if (a,b) and (a,b') are both in f, then b=b' (for each a in A there exist at most one b in B such that (a,b) is in f)
 
  • #3
would you be able to give an example of a cartesian product between two example sets called A and B so that I could see the resulting set from that?
 
  • #4
cam875 said:
would you be able to give an example of a cartesian product between two example sets called A and B so that I could see the resulting set from that?

{1, 2, 3} x {a, b, c} = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (3, a), (3, b), (3, c)}
 
  • #5
The set definitions of pairs, tuples, relations, functions, and integers only serves to show that set theory can be used as a formal basis for pretty much all of mathematics. The point of set theory was to reduce as many ideas as possible to as few as possible. It makes everything it can into a set.

If you have done any computer programming before, it's essentially the same concept. Computer languages only understand a limited set of objects: integers, floating point numbers, unicode characters, functions, and lists of other objects (or some similar-looking list). But then, how do you program a 3-D game? There are no "vectors" in the computer's eyes! The answer is you have to "build" vectors yourself out of what's given. A vector might be a list of floating point numbers, coupled with functions that can act on them (addition, scaling, dot products). What about a matrix? Well, now that we've defined vectors, a matrix is simply a list of vectors and functions for how to multiply them, transpose them, take their determinants, etc.

We're doing the same thing here with sets. You start off with only sets. Nothing but sets. Well, how do we make ordered pairs? We define them in terms of sets: (a, b) = {{a}, {a, b}} or whatnot. What about relations? We define a~b to be a set of pairs. What about functions? We define a function as a relationship with a condition that no two pairs (a, b) and (a', b') in the relation can have a = a'.

In a sense, we are merely "programming" the rest of mathematics with set theory. After we have defined a function in terms of sets, we can treat it as if functions are their own "real" entites. Once we have proven basic theorems about functions by their definition (composition is associative, and basic principles behind "onto" and "one-to-one" functions), we can more or less completely forget about the definition, knowing that as long as we defined it properly and made no mistakes in our core theorems, evereything else we prove with them will be correct, regardless of the details of the definition.

The take home message is, if sets are the only "primitive" object you have at your disposal, you can define many other mathematical objects in terms of them. Functions are not *really* sets. But if you went to a planet where functions were not yet discovered, but where the natives understood sets, you could give a precise description of what Earth mathematicians mean by "function."
 
  • #6
make sense and yes I am an avid programmer so it does help me understand with the analogy u gave me, thanks
 

Related to Functions and Sets: Get Help Understanding Them

What are functions and sets?

Functions and sets are mathematical concepts used to describe relationships between elements in a collection. A function maps each input element to a unique output element, while a set is a collection of distinct elements.

What is the difference between a function and a set?

The main difference between a function and a set is that a function describes a relationship between elements, while a set is simply a collection of elements. A function can be thought of as a special type of set where each input has only one corresponding output.

How do I represent a function or a set?

Functions and sets can be represented in various ways, such as using mathematical notation, diagrams, or tables. For functions, the most common notation is f(x) = y, where x is the input and y is the output. Sets can be represented using braces { } and listing the elements inside, or using set-builder notation {x | x satisfies a certain condition}.

What is the domain and range of a function?

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In other words, the domain is the set of x-values and the range is the set of y-values in a function.

How are functions and sets used in real life?

Functions and sets are used in various fields such as mathematics, physics, economics, and computer science. They can be used to model relationships between quantities, analyze data, and solve problems. For example, in economics, functions can be used to describe the relationship between supply and demand, while sets can be used to represent different categories of goods or services.

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