 So in the previous video, we just looked at these associative and commutative properties, but let's just let's just Define them properly and if we define them properly we can use them in future proof. So the first Type of binary operation. We're just going to talk about. Let's get a new and you pin The first one we're going to talk about is just this commutative property. So let's have a look commutative There we go. This commutative type of binary operation and what we're going to define that is on some set Let's define this first. Let's have our binary operator our binary operator and Let's make that just this generic symbol just our little circle. So we have our binary operator We have the set on which we going to This or it's going to apply to elements this operation on elements of the set and let's have this set X And let's have two elements of that set. We're going to have some elements of that set Elements and let's make it X1 and X2 and they are all both elements of the set And we say that this operation this binary operation on the elements of this set is commutative We as human beings define it that is our choice This is how we define it if the following property holds if I have X1 the binary operation on X2 That equals X2 binary operation X1 if I did this operation and I did that I would get exactly the same solution That is our choice. That is how we define it once we've defined it We can now use it when we go To look for proofs the same we're going to go if we look at the associative property Let's put it up here the associative associative property So this associative type of binary operator So our binary operator still is this going to be this is our binary operator Set is still going to be a set of elements And we call that set X and now the elements that we're going to have is we're going to have one Two and three elements They are all elements of our set X They are all elements and we say now that this binary operation on these elements of the set is Associative if the following holds again as human beings we decide this is our definition once we've defined it We can use it for instance in a proof and it is going to hold if the following is such that if I take these two elements Of X any arbitrary elements of X and I do the binary operation there I do that first and this result is then X3 I get exactly this That those would be exactly the same So we have to find this under these Circumstances we have to find this as an associative type of binary operator and in a commutative type of binary operator On elements of a certainty that is how we define those