Original subtitles

Hi guys and this lecture I will talk about relations over some said relation means that two elements

are in some relation to each other.

For example free is in the relation smaller than with five.

I am in the relation of being son with my dad.

I hope that makes sense and I think it does because we are in touch with all kinds of relations all

the time.

Now let's talk more about relations in mathematic terms relation on set A is some subset of a times.

So now let's define what a times a means.

In general we can multiply two different sets so let's say we want to multiply A and B and the result

is also said that it's containing pairs where the first part is from set A.

And the second part is from said be like this.

So for example if a contains A B C and B contains 0 and 1 a times b where is out into set containing

bears a and zero A and one B and zero B and one see ends zero and C and 1.

So if you want to calculate something like this you can just pick one element from a and then put him

together with all the elements from B and then you move to next element and.

And when you reach the end you are done.

So to get back to our example let's say that a contains number 1 2 and 2 free.

So we compute a times a.

The result of this operation looks like this.

And now we can create the relation over array and we can do that by picking berries from this set.

So for example if I have a relation that is defined for example like X minus XY is smaller or equal

to zero Overbay it will contain beris from set A times a.

So this relation will contain pairs 1 and 1 1 and 2 1 and 3 because 1 minus 1 is zero.

And we want the result to be smaller or equal to zero.

So that is correct 1 and 2 because one might assume is minus one.

And same with one and free back.

And this relation will not be two and one because two minus one will give us one and that is bigger

than zero but there will be pair containing 2 and 2 2 and free and free and free.

So these are all the pairs from set A that are in relation are.

Sometimes people write this fact like this.

So for example two are two.

That means two is in relation are with two or they write are inside parentheses two and 2.

That means the same thing.

Now you might ask.

And what is it good for.

Well you probably know how data base looks like that.

They didn't know that.

We have something called relational database model.

And in this model rows in database are elements of some relation.

So now I showed you example of something called binary relation and this means there are two elements.

And during this relation.

So it is a relation between x and y.

But we can have a relation between and elements where n is some natural number.

That way we are picking elements from a time say times a times a where a is there and times.

So for example if this free relation R can look like X is smaller than I and II is greater than z that

is just one of many examples and this relation will contain 1 2 1 1 3 1 2 free 2.

I hope you understand it.

Now let's talk about properties of binary relations.

So first property is reflexivity and what that means.

Well if we have a relation overlay this relation is reflexive when every element for May is in the relation

with itself.

So if you take a look at our first example it is reflexive because one is in relation with one two is

in relation with two and three is a new relation with free.

I hope that makes sense.

Every element from a MUST be in relation with itself.

Next property is symmetry.

And what that means.

Well in general if x and y is in relation I and X must also be in relation.

So if we look at our relation it is obviously not symmetric.

For example 1 and 3 is in relation dot free and Dujuan is not.

So in order for a relation to be symmetric there must be a kind of opposite pairs next one is antisymmetry.

And what that means when some mix and is in the relation and I and x is in the relation also then x

is equal to II and if we look at this relation it is anti-symmetric because the only pairs that are

symmetric are 1 and 1 2 and 2 and free and free y let's say that x and y is 1 and 1 then we need to

find in this relation I and X but since I and X are equal we are trying to find one and one and one

and one is obviously in this relation.

Similarly with our bears and for all of these pairs it's also true that X is equal to II.

So that means this relation is anti-symmetric OK.

Let's look at last Twan transitivity relation is transitive when for every x and y and z from a.

It's true that if x and y is in relation and I and Z is in the relation then x and z is also in relation.

So once again let's take a look at our example.

It is transitive.

I am sure you're not seeing it right away.

And that is OK.

This takes practice like anything bad.

If I show you example if I pick one and two as a x and die.

Now I am looking for some bear that starts with two.

So I can for example pick two and three and then I need to find one and three in this relation since

one is our X and free is our z and I am able to find one and free in this relation.

But this must be true for every pair.

So you can try to check the rest on your own.

And with that being said if you have any questions just ask and I will see you next time.