Introduction to Sets
by mrichard, Oct 30, 2018, 4:08 AM
Preface to this Blog
Quite a while ago, an endeavoring individual tried to start an open-source repository of mathematical information called Mathbook. I contributed an article, which I'll put down in two parts on this blog as a start.
It seems that the project has died, although the website is still available. While this is a bit of a shame, I would like to give some of my own little lessons here. The creator's idea behind Mathbook was to focus on giving people an understanding of why we do math in a certain way. This is missing from mathematical curriculum today, but it is vital to understand that when math was developed, decisions were made for specific reasons.
In particular, while AoPS regulars probably don't concern themselves with these questions (they're pretty intrinsically motivated), the general populace and self-dubbed "Non Math People" do not have the same appreciation for mathematics. Efforts to spice up curriculum with applied questions always leads to contrived problems that, to me, really miss the entire point of mathematics.
Yes, mathematics is a very deductive, rational field. But it is also incredibly creative and inventive. Math is its own reward, its own justification (as discussed by Hardy in his Mathematician's Apology.) I want people to see some of the cool stuff that happens in math, but also build stuff up from their basics. I'll get a lot wrong, miss some explanations here and there, but I hope people will come along for the ride every once in the while. This will be a long shot from the Infinite Napkin, partially because I don't have the encyclopedic knowledge of Evan, but it will hopefully fill a small gap I see in how mathematics is viewed on a wider scale, that is generally not addressed by AoPS.
Introduction to Sets
We'll learn the basics of how sets are used in mathematics. It is important to understand basic arithmetic before diving in, but nothing else really. Just an openness of thought.
Understanding sets
In any field of mathematics, it is important to be able to deal with objects and structures. At the lowest level of mathematical objects and structures are sets. (Of course, one could dive into categories as a more general "basic" object, but we'll stick with the sets.)
Most simply, a set is a collection of objects. We can think of the set of all flowers in Hawaii, or the set of whole numbers between 10 and 37. Typically, we use curly brackets (braces) to denote a set, such as
If we are using the same set many times in a row, or talking about a set that cannot easily be written down, we can use some other symbol. Throughout this post we will let 
be the set 
and 
be the set of all flowers in Hawaii.
There are certain rules and terms used with sets that allow mathematicians to be consistent when using and talking about sets. For example, we want to know what to call the objects in our sets, and how we can write such a relationship down.
It is also natural to discuss how many elements are in a set.
For this tutorial, we will only be looking at sets with finite cardinality; this means we will always be able to list and count every element in the set. Future posts will explore larger sets, which becomes an even more powerful (and fun!) mathematical tool.
Often we want to look at some of the elements in a set, but not all of them. For example, we might want the elements of which are red flowers. This is a very common notion in mathematics: given an object or structure, how can we look at smaller objects that have a similar structure?
When doing mathematics, it is good practice to look at the simplest example of any object you are interested in exploring. When it comes to sets, it becomes natural to ask "What if my set has no elements?"
Almost everything you see and do has sets hiding in the background. They are a universal way of communicating mathematical ideas and structures, and are thus very important to understand.
Questions to Ponder
Quite a while ago, an endeavoring individual tried to start an open-source repository of mathematical information called Mathbook. I contributed an article, which I'll put down in two parts on this blog as a start.
It seems that the project has died, although the website is still available. While this is a bit of a shame, I would like to give some of my own little lessons here. The creator's idea behind Mathbook was to focus on giving people an understanding of why we do math in a certain way. This is missing from mathematical curriculum today, but it is vital to understand that when math was developed, decisions were made for specific reasons.
In particular, while AoPS regulars probably don't concern themselves with these questions (they're pretty intrinsically motivated), the general populace and self-dubbed "Non Math People" do not have the same appreciation for mathematics. Efforts to spice up curriculum with applied questions always leads to contrived problems that, to me, really miss the entire point of mathematics.
Yes, mathematics is a very deductive, rational field. But it is also incredibly creative and inventive. Math is its own reward, its own justification (as discussed by Hardy in his Mathematician's Apology.) I want people to see some of the cool stuff that happens in math, but also build stuff up from their basics. I'll get a lot wrong, miss some explanations here and there, but I hope people will come along for the ride every once in the while. This will be a long shot from the Infinite Napkin, partially because I don't have the encyclopedic knowledge of Evan, but it will hopefully fill a small gap I see in how mathematics is viewed on a wider scale, that is generally not addressed by AoPS.
Introduction to Sets
We'll learn the basics of how sets are used in mathematics. It is important to understand basic arithmetic before diving in, but nothing else really. Just an openness of thought.
Understanding sets
In any field of mathematics, it is important to be able to deal with objects and structures. At the lowest level of mathematical objects and structures are sets. (Of course, one could dive into categories as a more general "basic" object, but we'll stick with the sets.)
Most simply, a set is a collection of objects. We can think of the set of all flowers in Hawaii, or the set of whole numbers between 10 and 37. Typically, we use curly brackets (braces) to denote a set, such as
If we are using the same set many times in a row, or talking about a set that cannot easily be written down, we can use some other symbol. Throughout this post we will let 
be the set 
and 
be the set of all flowers in Hawaii.There are certain rules and terms used with sets that allow mathematicians to be consistent when using and talking about sets. For example, we want to know what to call the objects in our sets, and how we can write such a relationship down.
to be a separate set. We can either just call this an invalid set, or combine the 
s
is in the set 
It is also natural to discuss how many elements are in a set.
For this tutorial, we will only be looking at sets with finite cardinality; this means we will always be able to list and count every element in the set. Future posts will explore larger sets, which becomes an even more powerful (and fun!) mathematical tool.
Often we want to look at some of the elements in a set, but not all of them. For example, we might want the elements of
which are red flowers. This is a very common notion in mathematics: given an object or structure, how can we look at smaller objects that have a similar structure?
and 
If every element in 
,
For example, the set 
is a subset of 
When doing mathematics, it is good practice to look at the simplest example of any object you are interested in exploring. When it comes to sets, it becomes natural to ask "What if my set has no elements?"
,
,Almost everything you see and do has sets hiding in the background. They are a universal way of communicating mathematical ideas and structures, and are thus very important to understand.
Questions to Ponder
By definition, this means 
to mean 
for any type of subset, and specify 
to mean 
July 2019
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