The first tip is to stake Hung.

People often hear the word mathematics and panic, and when you panic, you stop listening.

And if you stop listening, you can't learn the material.

So if you simply stay calm, then you will be able to learn the material better.

You aren't being graded on the subject, so there's no need to panic.

The second is to rewind.

Uh, you can rewind the videos at any time.

And when I learned discrete mathematics, I didn't have this opportunity.

So in theory, you should be able to learn the material much faster than I was ever able to.

So I strongly encourage rewinding videos, Finally explained.

Once you think you understand the material of videos, you can try to explain the material out loud, either to yourself or to a friend.

If no one's around or you feel we're talking himself, then you can try to explain it to a rubber ducky.

There's a concept in programming called rubber duck debugging, which is where programmers go through their code line by line, trying to identify bugs in their code by explaining it to rubber ducky.

So, similarly, if you go through the material of the videos subject by subject, you will soon be able to identify uh, any gaps in your understanding of the material.

In this video, I'll be explaining what discrete mathematics is and why it is important for the field of computer science and programming.

The street mathematics is a branch of mathematics that deals with discreet or finite sets of elements rather than continuous or infinite sense of elements.

Imagine trying to run a program that requires an infinite number of executions to complete attacks.

It's obvious to say that the program was run forever and the task would never be completed because they're an infinite number of executions.

In order.

Avoid this problem.

We approximate to continuous sets with discrete sets.

Now you may be thinking, I never used math that involves infinite sets, but I promise that you do.

The simplest example is with a circle.

A circle.

By definition, is an infinite number of points equidistant fixed, and the problem with this is that if we try to write a program that prints out all of these points.

It will run forever because they're an infinite number of points and therefore an infinite number of executions.

So this is physically impossible.

That's why if we zoom in here, you can see that when you come down here, they're all these points.

But in reality, we should have even more points between these points.

And then if we zoom in on those, we'll have more points between those points and we can never complete.

The task now involves seeing circles on computers.

So what's going on?

How is this possible?

We just established that it's impossible.

Well, the answer is that they're approximations.

For example, consider regular polygons regular polygons, like a triangle or square or a Pentagon.

Those don't really look like circles, but if you keep increasing the number of vergis ese, uh, eventually you'll start getting hexagons.

Octagon Decade guns, hex a decade guns, I Costa guns.

You can see that these regular polygons, the Maur more.

You keep increasing the number of urgencies which the verte Cesaire equidistant from a fixed point.

They will eventually approximate a certain.

Eventually there will be indistinguishable to the naked eye and we'll look identical to a circle.

And in this video I'll be explaining what sets are.

A set is a collection of distinct objects called elements or members.

An element is essentially anything you wanted to be.

And for example, elements could be numbers, letters, variables, more sets or nothing at all.

Sets are usually denoted by capital letters such as capital, a capital, B, C, E or F, and we say that the set contains elements.

For example, we could say that the set A equals the set containing the Element five, or we can say that five is an element of a Now I will be interesting.

A lot of symbols like this, this just means is an element of and it'll save us a lot of time in future videos and I don't have to write out is an element of each time, so that'll come in handy.

So what does this set actually look like?

Well, there are two common types of notation for sets, and the one I'm introducing in this video is roster temptation.

But it looks like this is just curly braces with elements inside separated by commas, which is very similar to Java script, The only difference being that in math we use curly braces and JavaScript.

We use a square bread bracket.

So, uh, now let's just walk through these.

So the the set A equals the set containing the album.

Five.

That's how you would read this.

The set B equals the second tainting elements 23 and four.

The set C equals the set containing elements D E F G now dfmgi They're not.

They don't have to be variables.

They're either variables or they could be characters.

It just depends on the context.

The set e It contains no elements in this case, which has an impact that you'll see in future videos.

And finally, the set f equals the second Training elements A, B and C in this case are elements are sets.

So we would read this as f equals the set containing the set containing the Element five, the set containing elements 23 and four and the set containing Dfmgi.

So in the next video, I'll be going over a different form of sanitation in addition to introducing you to some common sense, and this will set us up eventually.

My goal is to bring us up to being able to define and understand algorithms, and this is where we have to begin.

In this video, I'll be introducing interval notation and talk about some common sets.

This will lay the foundation for the next video on set builder notation.

The beauty of interval notation is that allows us to officially describe all numbers between two values.

Suppose we want to say that the variable X is between zero and one.

Well, we could do just that.

We could say that X is an element of zero.

The interval 01 or which is really saying X is greater than zero in X is less than one.

Likewise, if we wanted to say that X is greater than or equal to zero and X is less than one, then we have to switch this to a square bracket.

All we're doing is we're now including the number zero in the interval.

Also, we can include zero, and we can also include one in the honorable by using square brackets on both sides.

So it's greater than or equal to less than, uh and now talk about some common sets.

The 1st 1 being the no set, which is the same thing, is the empty set, which is just There are no elements contained within the set, and this will come in handy in the future.

The funny And here is the natural numbers, which is the set containing elements 123 and then these three dots here called an ellipsis, which tells us that the pattern continues.

So N is the set containing 1234567 alway off to infinity.

This funny end with zero is the natural numbers.

However, it includes the value zero within the set.

So it 012 and then the ellipsis tells us all the way to infinity and an energy fashion.

0123456 On Finally, we have all of the energy is positive and negative and not which is zero negative one native to although down negative infinity and 01234 all the way up to infinity.

In this video, I'll be introducing the rational numbers and settled in rotation.

The rational numbers are defined as a ratio of to emitters, and examples include the images themselves terminating decimals and repeating decimals So let's start with repeating decimals.

If we let 10 X equal.

9.999 Repeating then If we divide by both sides by 10 then X equals 0.999 Repeating.

And if we subtract x from 10 x, we have nine x equals not and then divide both sides.

By nine we have X equals one.

And then, if we multiply by 10 both sides by Tammy of 10.

X equals 10.

So now we have 10.

X equals 10 but 10.

It's also equals 9.99 repeating.

So we have now.

And these are in fact true.

9.99 Repeating equals 10.

So we think we can now express 10 X as an imager, which is a a, um, rational number.

We can put 10 divided by one, which is ratio.

So another problem we have is terminating decimals.

If we have 2.78 equal, why, then we can just easily say 278 divided by 100 equals.

Why, as well these air to imagers and it's a another rational number.

So a convenient way to define these there is denoted with a Q and This is called set builder notation.

We have the Q equals the sack containing elements a divided by be such that A and B are elements of the inner jurors with the only condition that be cannot equal zero.

If you divide by zero, it sze undefined.

It doesn't mean anything.

So if you're interested in that, I recommend Googling and I think it's interesting in this video will be giving an example of a non rational number.

If the square to were rational number, then by definition it must be expressed as the ratio of two integers.

If we then square both sides, we have two equals, a squared divided by B squared.

If we multiply, both sides might be square.

We have to be squared, equals a squared.

And from this, we know that a square is even a score's, even because we have two times some imager, which is be square.

Since a score's even, we know that a is even because a squared equals two times a manager.

The energy I chose was to K square.

That means that any times they could be expresses to K Times two k, which is to say that a equals two K.

I can now substitute two times two K square for a square, and that leaves us with B squared equals two K square.

Once these twos cancel out.

Now, this also means that B squared is even because B squared equals two times a manager.

Which is to say that B is also even following this same logic.

Now, if we go back to our original premise, uh, the square to two equals a divided by B, where a divided by B isn't irreducible fraction.

But since A is in even number and B is also uneven number, we can express these numbers as to K in Tooele.

But these twos here will reduce meaning that we'll have the square to two equals k divided by m.

But since this reduced, since this is no longer explicitly represented as a devoted might be and it reduced to Kate about abi l, we know that we do not have a rational number.

Now.

There's a more specific way to define these numbers or classify these numbers, but we need set operators in the next video.

In this video, I'll be defining four binary operators for sex.

I've defined two sets A and B teau help give some examples.

So the first is a union be, which equals a set containing elements p such that pees an element of A or P is an element of B.

The next is the intersection.

A intersection B equals the second training elements.

P such that pees an element of A and P is not gonna be the next is the set difference of A and B, which equals the set containing P or elements.

P such that pees an element of A and P is non development of beat.

And finally, we have the symmetric difference, which is I'm gonna be very helpful when you actually complete the symmetric difference algorithm challenge.

So the symmetric difference of A and B equals the set difference of a and B union with the set difference of being a and finally I will define the irrational numbers which I hinted to at the end of last video.

So you're rational numbers are simply the rial number, real numbers minus the rational numbers, and that equals the irrational numbers.

In this video, I will be using Venn diagrams to give a graphical representation of the Sep operators we learned in the previous video.

So if you recall the definition of a union, be you know that a union be equals the set containing elements ex such that X is an element of a or X is an element of beat.

So if we let this circle represent the set A and we let this circle represent the set B, then that means that X can land anywhere in any of these circles.

Which is why this rent the entire region is shaded red were saying that this river region represents all possible values of X.

And if we were called a definition bait of a intersection beat that equals the set containing elements ex such that X is an element of A and X is now gonna be.

Which is why the overlap where a and B overlap is shaded red were saying all possible values of eggs.

This red region Now the set difference of A and B equals the set containing elements ex such that X is an element of A and X is also not an element of be, which is why it's just this region here.

None of the elements air within be the set difference of being A is the set containing elements ex such that X is an element of B and X is not an element of a and finally, the symmetric difference of A and B I defined as the union of a Minus B and B minus A.

So we take this region in this region and we take the union of it, which is why this is the resulting region.

There's no overlap between the two now.

This is a really good way to familiarize yourself with the concept of set operators.

However, don't rely on you should really focus on the definitions as it will help you out in the long term.

In this video, I'll be introducing the concept of subsets in super sense.

The dots on the board here represent elements of a set.

The lines represent the sets themselves.

These lines have been labeled B a u.

We can say that B is a subset of a because be contains because all elements of B are elements of A.

We can also say that B is a proper subset of a because all elements of B R elements of A and there are elements of a that are not within the set B.

We can also say that the set A is a super set of set B because A contains all of the elements of B and being even more specific, we can say that is a proper super set of B because a contains all elements of B and there are elements of a they're not elements of B.

What we cannot say is that you is a proper subset of a Although this circle here is bigger than a like it just takes up more space.

It doesn't contain any more elements.

So all the elements So you contains all the elements of a however you does not contain any extra elements.

So you is a subset of a and A is actually a subset of you.

You was a super set of a and is also a super set of you.

So he sets a and you, in this case, are actually equal.

In this video, I'll be introducing the concept of the universal set and compliments.