 So in the section, let's start talking about what we're going to term relations relations Now that builds on something that we've seen before and that was the product set and Relation is just a subset of the product set of a set onto itself So I'm going to have this set a and its product set just a With itself and I'm going to call my relation. I'm going to use this uppercase R It's just going to be my relation Now let's have a quick example because that will explain really well what a relation is so imagine my my my set a is just the values one two three four five and six and my relation is my relation could just be Divide Just make divides So that looks a bit odd, but let's have a look at this Then we could also see the notation that we are going to use does one divide itself indeed it does so one divides one One R1 now they are both in the set It is a product set and we could use product set notation that we used before Just in a little bit after I'll show you this does one divide two it does not divide three four five or six with art Remained this two divide itself. Yes, so two divides two Two also divides four two are four and two divides six and three divides itself and three also divides three also divide six and Four does not divide six four divides itself Five divides itself and six divides itself lots of squeaking from this almost right pin today so this will be the relations of All the elements of this relation divides. It's a subset of this product set Which means that we can really write this in two ways now if we make this a One and a two we can write a sub one comma a sub two or a sub two comma a sub one And you see many textbooks are going to use the product set notation. So we actually here going to have that are let's get a new pen That are my relation It's really just going to equal the following. It's going to be one comma one and then two comma two and Then four four comma two so four comma two And it's going to have six comma two and it's going to have three comma three Three and three and it's going to have six and three It's going to have four and four and it's going to have five and five It's going to have six and six and that is my set So I'm using product set notation here But you see it is just the other way around of this notation Just check sometimes some do use the notation two comma four instead of four comma two So just make sure which one you are using. So this is a relation. Let's look at one more relation Let's have my set a My set a is the set of all elements x such that let's make that one is Less than or equal to x which is less than or equal to 20 and I have the fact that x is an element of the set of natural numbers one two three four five until 20 and my relation is is three times are is three times So that might sound a bit odd again like that one. So are is this going to be the following? so three is three times one Let's have six is three times two Let's have six Let's have eight is three times. So three times. He's a nine. It's going to be three times three 12 is three times four 15 is three times five 18 is three times six so there we have all the relations of This set the price is a subset of the product set of A unto itself and I can write it Just in product set notation. So once again, we're gonna have one and three One and three we're gonna have two and six We're gonna have three and nine and we're going to have four and twelve And we're gonna have five and fifteen and we're gonna have six and eighteen So we have those pairs. So that is a binary relation. We are dealing just with two Just from A unto itself. So the product set A A and A and we have this binary relation next up We're going to just look at some properties of these relations