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Hi guys and this lecture I will talk about relations over some said relation means that two elements
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are in some relation to each other.
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For example free is in the relation smaller than with five.
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I am in the relation of being son with my dad.
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I hope that makes sense and I think it does because we are in touch with all kinds of relations all
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the time.
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Now let's talk more about relations in mathematic terms relation on set A is some subset of a times.
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So now let's define what a times a means.
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In general we can multiply two different sets so let's say we want to multiply A and B and the result
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is also said that it's containing pairs where the first part is from set A.
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And the second part is from said be like this.
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So for example if a contains A B C and B contains 0 and 1 a times b where is out into set containing
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bears a and zero A and one B and zero B and one see ends zero and C and 1.
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So if you want to calculate something like this you can just pick one element from a and then put him
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together with all the elements from B and then you move to next element and.
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And when you reach the end you are done.
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So to get back to our example let's say that a contains number 1 2 and 2 free.
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So we compute a times a.
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The result of this operation looks like this.
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And now we can create the relation over array and we can do that by picking berries from this set.
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So for example if I have a relation that is defined for example like X minus XY is smaller or equal
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to zero Overbay it will contain beris from set A times a.
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So this relation will contain pairs 1 and 1 1 and 2 1 and 3 because 1 minus 1 is zero.
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And we want the result to be smaller or equal to zero.
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So that is correct 1 and 2 because one might assume is minus one.
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And same with one and free back.
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And this relation will not be two and one because two minus one will give us one and that is bigger
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than zero but there will be pair containing 2 and 2 2 and free and free and free.
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So these are all the pairs from set A that are in relation are.
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Sometimes people write this fact like this.
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So for example two are two.
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That means two is in relation are with two or they write are inside parentheses two and 2.
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That means the same thing.
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Now you might ask.
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And what is it good for.
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Well you probably know how data base looks like that.
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They didn't know that.
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We have something called relational database model.
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And in this model rows in database are elements of some relation.
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So now I showed you example of something called binary relation and this means there are two elements.
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And during this relation.
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So it is a relation between x and y.
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But we can have a relation between and elements where n is some natural number.
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That way we are picking elements from a time say times a times a where a is there and times.
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So for example if this free relation R can look like X is smaller than I and II is greater than z that
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is just one of many examples and this relation will contain 1 2 1 1 3 1 2 free 2.
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I hope you understand it.
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Now let's talk about properties of binary relations.
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So first property is reflexivity and what that means.
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Well if we have a relation overlay this relation is reflexive when every element for May is in the relation
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with itself.
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So if you take a look at our first example it is reflexive because one is in relation with one two is
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in relation with two and three is a new relation with free.
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I hope that makes sense.
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Every element from a MUST be in relation with itself.
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Next property is symmetry.
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And what that means.
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Well in general if x and y is in relation I and X must also be in relation.
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So if we look at our relation it is obviously not symmetric.
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For example 1 and 3 is in relation dot free and Dujuan is not.
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So in order for a relation to be symmetric there must be a kind of opposite pairs next one is antisymmetry.
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And what that means when some mix and is in the relation and I and x is in the relation also then x
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is equal to II and if we look at this relation it is anti-symmetric because the only pairs that are
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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
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find in this relation I and X but since I and X are equal we are trying to find one and one and one
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and one is obviously in this relation.
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Similarly with our bears and for all of these pairs it's also true that X is equal to II.
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So that means this relation is anti-symmetric OK.
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Let's look at last Twan transitivity relation is transitive when for every x and y and z from a.
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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.
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So once again let's take a look at our example.
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It is transitive.
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I am sure you're not seeing it right away.
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And that is OK.
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This takes practice like anything bad.
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If I show you example if I pick one and two as a x and die.
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Now I am looking for some bear that starts with two.
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So I can for example pick two and three and then I need to find one and three in this relation since
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one is our X and free is our z and I am able to find one and free in this relation.
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But this must be true for every pair.
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So you can try to check the rest on your own.
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And with that being said if you have any questions just ask and I will see you next time.
8339
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