It is a good time to introduce, a Convenient Language.

We will use language of sets.

Set is a very simple objective.

It's an arbitrary group of arbitrary elements, arbitrary objects.

So we will denote sets by capital letters A,

B, C, S, and, so on.

And sets can be given,

for example, by list and all of their elements.

Sets are given by their elements,

and we can do, to define some set,

you can just give a list of elements, for example,

the set consists of four elements zero, one, two, three.

In this notation, order doesn't matter.

The order of elements in the set doesn't matter so we

can give the same set by the list zero,

one, two, three and by the list two,

zero, three, one. It's the same set.

And we allow repetitions in the list when we give a set in

this way so each element either belongs to the set,

or doesn't, so it can belongs there twice.

So the set one, zero, one, three, two, three,

the set of these elements is the same as just,

the set of elements zero, one, two, three.

So each element are belongs to the set, or not.

For us sets provide a convenient language.

In Mathematics, sets are very important,

they play fundamental role.

They are in the foundation of mathematics.

For us sets can be just any list of any elements,

for example this is set for us zero,

square root of two.

Isaac Newton, a leprechaun.

This set consists of four elements.

Here is the list of them.

So this is illegitimate set for us.

So in Mathematics, it is known that there are pitfalls there,

so you cannot just consider anything you want.

For example, the construction like this,

set consisting of all sets.

It is a dangerous construction.

You should be very careful with it.

So, but we will not encounter these difficulties in this course,

and we will not discuss,

and we will not have this problem, so, for us,

set is arbitrary group of arbitrary elements.

Okay, so the convenient way to view sets is by the means of Venn diagrams.

So here is a Venn Diagram for two sets,

and here there are two sets A and B.

Elements of A are depicted within the left circle.

Elements of B are depicted within the right circle.

So note of the circles are intersecting,

and intersections correspond to elements that belongs to both sets,

so are [inaudible] elements that are in both sets will be in the intersection of two circles.

So suppose we have two sets,

and here is a picture,

and we would like to introduce a couple of operation over sets.

First the intersection of two sets.

And the intersection of two sets is again a set,

and these set of all elements,

that belong to both A and B.

Next separation is a union of two sets, and these is a sets.

It consists of all elements that belong to either A, or B.

If you have seen our first course what does it proof?

Note that I recall that we have,

we will declare under variations,

and, or on Boolean variables,

and note that the operations for intersection,

and union are similar in their looks by two operations,

or in Boolean logic.

And I recall also that we have Venn diagrams there as well,

and the Venn diagram for and operation looks like that,

like for intersection in the Venn diagram,

or operations looks like,

the Venn Diagram, for Union.

This is not against,

There is various correspondence between boolean operations and operations or sets.

We will discuss it in details,

but there is a connection,

there is a correspondence.

Okay, and the last definition that we would like to introduce is,

the number of elements in the set.

Which you know like this,

note that the set can be infinite,

for example, the a set of all natural numbers.

It is an infinite set,

so this number of elements in the set can be infinite.

It can also be zero.

There is a set of no elements, that are,

so-called empty set, so this is our last notation here.

Okay, and now we can proceed to, back to Combinatorics.