Sets n Vein Diagram(introduction of set)

 

Definition

What is a set? For make it easier to understand me make it simple, set just like collection. We need to specify common property among the things and we gather up all the things that have this common property.

First we need to specify a common property of the "things" (this word will be defined later) and then we gather up all the "things" that have this common property.

 Example of set the items you wear: shoes, socks, hat, shirt, pants, and so on. It just been grouped together.

Notation

. We simply list each element (or "member") separated by a comma, and then put some curly brackets around the whole thing:

Simply we list each element separated by comma and we put some curly bracket around it. This is the curly bracket “ { } “ or called “sets brackets”.

This is the notation for the two previous examples:

{socks, shoes, watches, shirts, ...}

Numerical Sets

If we already define a set, first we need to specify the  common characteristic. We also can do sets in number. Set of even numbers: {..., -4, -2, 0, 2, 4, ...}

This Set of odd numbers: {..., -3, -1, 1, 3, ...}

This Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}

The Positive multiples of 3 that are less than 10: {3, 6, 9}

There also can be sets of numbers that not have common property, we just need to defined the way for example.

{1, 3, 6, 424, 3639, 4427}

{4, 5, 6, 5, 21}

{2, 929, 46282, 42442659, 119244203}

all sets that I just randomly banged on my keyboard to produce.

Universal Set

				We used word “thing” and it called universal set. Set is include integers. The universal set would be integers.

More Notation

We commonly use capital letter to represent the sets and lowercase letters to represent the element in the sets. for example, B is b set, and a is an element in B. Same with c and c, and D and d.

Now you don't have to listen to the standard, you can use something like m to represent a set without breaking any mathematical laws (watch out, you can get π years in math jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not?

Also, when we say an element b is in a set b, we use the symbol to show it.

And if something is not in a set use .

Example: Set A is {1,2,3}. We can see that 1 A, but 5 A

Equality

Two sets are equal if they have the same members. Now, at first glance they may not seem equal, so we may have to examine them closely!

Example: Are b and C equal where:

• B is the set whose members are the first four positive whole numbers
• C = {4, 2, 1, 3}

Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so: Yes, they are equal!

And the equals sign (=) is used to show equality, so we write:

A = B

this the video for set:

 

 

         Introduction to Sets

Subsets

When we define a set, if we take pieces of that set, we can form what is called a subset.

So for example, we have the set {1, 2, 3, 4, and 5}. A subset of this is {1, 2, and 3}. Another subset is {3, 4} or even another, {1}. However, {1, 6} is not a subset, since it contains an element (6) which is not in the parent set. In general:

A is a subset of B if and only if every element of A is in B.

So let's use this definition in some examples.

Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?

1 is in A, and 1 is in B as well. So far so good.

3 is in A and 3 is also in B.

4 is in A, and 4 is in B.

That's all the elements of A, and every single one is in B, so we're done.

Yes, A is a subset of B

Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A.

Let's try a harder example.

Example: Let A be all multiples of 4 and B be all multiples of 2. Is A a subset of B? And is B a subset of A?

We can’t check every element of set. We need to get an idea of what the element look like in each then we compare them all.

The sets are:

• A = {..., -8, -4, 0, 4, 8, ...}
• B = {..., -8, -6, -4, -2, 0, 2, 4, 6, 8, ...}

By pairing off members of the two sets, we can see that every member of A is also a member of B, but every member of B is not a member of A:

 

So:

A is a subset of B, but B is not a subset of A

Proper Subsets

we need to look at the definition of subset.

Let A be a set. Is every element in A an element in A? (Yes, I wrote that correctly.)

Well, umm, yes of course, right?

So doesn't that mean that A is a subset of A?

This seen not proper yet, so we change the subset to be proper using proper subsets

A is a proper subset of B if and only if every element in A is also in B, and there exists at least one element in B that is not in A.

This little piece at the end is only there to make sure that A is not a proper subset of itself. Otherwise, a proper subset is exactly the same as a normal subset.

Example:

{1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}.

Example:

{1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set.

Notice that if A is a proper subset of B, then it is also a subset of B.

Even More Notation

When we say that A is a subset of B, we write A B.

Or we can say that A is not a subset of B by A B ("A is not a subset of B")

When we talk about proper subsets, we take out the line underneath and so it becomes A B or if we want to say the opposite, A B.

Empty (or Null) Set

This is little bit weird and also make little bit confuse.

 

As an example, think of the set of piano keys on a guitar.

"But wait!" you say, "There are no piano keys on a guitar!"

And right you are. It is a set with no elements.

This is known as the Empty Set (or Null Set).There aren't any elements in it. Not one. Zero.

It is represented by

Or by {} (a set with no elements)

Some other examples of the empty set are the set of countries south of the south pole.

So what's so weird about the empty set? Well, that part comes next……

Empty Set and Subsets

So l back to our definition of the subsets. For example,we have a set A. We won't define it any more than that, it could be any set. Is the empty set a subset of A?

Going back to our definition of subsets, if every element in the empty set is also in A, then the empty set is a subset of A. But what if we have no elements?

It takes an introduction to logic to understand this, but this statement is one that is "vacuously" or "trivially" true.

A good way to think about it is: we can't find any elements in the empty set that aren't in A, so it must be that all elements in the empty set are in A.

So the answer to the posed question is a resounding yes.

The empty set is a subset of every set, including the empty set itself.

Order

No, not the order of the elements. In sets it does not matter what order the elements are in.

Example: {1, 2, 3, 4} is the same set as {3, 1, 4, 2}

When we say "order" in sets we mean the size of the set.

Just as there are finite and infinite sets, each has finite and infinite order.

For finite sets, we represent the order by a number, the number of elements.

Example, {10, 20, 30, 40} has an order of 4.

For infinite sets, all we can say is that the order is infinite. Oddly enough, we can say with sets that some infinities are larger than others, but this is a more advanced topic in sets.