Kid Exam: cool math for kids

Showing posts with label cool math for kids. Show all posts

Wednesday, 17 April 2013

cool math 4 kids

Cool math empty set and subsets

So let's go back to our definition of subsets. 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.

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Empty or null sets of cool math

cool math for kids empty or null sets

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.

proper subsets of kids cool math

Cool math proper subsets

If we look at the defintion of subsets and let our mind wander a bit, we come to a weird conclusion.

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 wouldn't that mean that A is a subset of A?

This doesn't seem very proper, does it? We want our subsets to be proper. So we introduce (what else but) 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.

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

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Tuesday, 16 April 2013

cool math Sets Equality

Sets equality in cool math for kids

In cool math 4 kids Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, you may have to examine them closely!

Example: A and B are equal where:

A is the set whose members are the first four positive whole numbers
B = {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!

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

A = B

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Sunday, 14 April 2013

Universal sets like cool math games

Universal Sets of cool math for kids

A set that contains everything. Well, not exactly everything. Everything that is relevant to the problem you have.

So far, all I've been giving you in sets are integers. So the universal set for all of this discussion could be said to be integers. In fact, when doing Number Theory, this is almost always what the universal set is, as Number Theory is simply the study of integers.

However in Calculus (also known as real analysis), the universal set is almost always the real numbers. And in complex analysis, you guessed it, the universal set is the complex numbers.

More Notation:

Also, when we say an element a is in a set A, we use the symbol to show it.
And if something is not in a set use .

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

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cool math games

Numerical Sets

So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers?

Set of even numbers: {..., -4, -2, 0, 2, 4, ...}
Set of odd numbers: {..., -3, -1, 1, 3, ...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Positive multiples of 3 that are less than 10: {3, 6, 9}

And the list goes on. We can come up with all different types of sets.

There can also be sets of numbers that have no common property, they are just defined that way. For example:

{2, 3, 6, 828, 3839, 8827}
{4, 5, 6, 10, 21}
{2, 949, 48282, 42882959, 119484203}

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

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Friday, 12 April 2013

sets like math games

Sets Notation

There is a fairly simple notation for sets. You simply list each element, separated by a comma, and then put some curly brackets around the whole thing.

The curly brackets { } are sometimes called "set brackets" or "braces".

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

{index, middle, ring, pinky}

The three dots ... are called an ellipsis, and mean "continue on".

Introduction of cool math Sets

Importance of Sets

In cool math 4 kids, Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when you apply sets in different situations do they become the powerful building block of mathematics that they are.

Math can get amazingly complicated quite fast. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number Theory, and the list goes on. But there is one thing that all of these share in common: Sets.

Example of Sets:

For example, the items you wear: these would include shoes, socks, hat, shirt, pants, and so on.

This is known as a set.

another example would be types of fingers. This set would include index, middle, ring, and pinky. So it is just things grouped together with a certain property in common.