 Next we're going to define certain types of elements. Some of the elements in a set might hold special properties and the first one we're going to look at is the identity element, identity element. So the identity element is going to be an element of a set. Let's make our set our set is the set of all elements x. We have a binary operation let's say a binary operation and a binary operation is just a little generic one that we used to now just a little x and we're going to have an element just an arbitrary element let's make it x sub 1 is an element of x and we are going to say that we now have this identity element we're going to call this our identity element u is an element of x if the following property holds. If I take u and now a binary operation with x1 and that commutes so it's got to commute as well x1 and u and I just get back x1 so no matter what element I take some arbitrary element if the binary operation under this commutative type of binary operation if I have this equal to each other and I just get back my original element if this holds then this u which is an element of x is a binary is an identity element a special element in my set and if it exists it's actually unique and we're going to look at the at the proof of that so let's look at an example let's look at the set of say for instance natural and let's say the set of integers if I look at the set of integers so zero is in there and I look at my binary operation of addition so I'm looking at my set of integers under addition so my set of integers under addition I have this unique identity element and that is the element zero because I can take any element inside of z let's make it z1 if I add to that zero that's the same as adding zero to that element and I get just get back that element if I look at the integers under multiplication so let's look at the integers under multiplication under multiplication and then my identity element of course is going to be is going to be 1 because z sub 1 times 1 equals 1 times z sub 1 and I just get z sub 1 back so that would be a special type of identity element but it is on a set under some binary operation with this binary operation must have this commutative property that's how we define it and I claim that if it exists it is unique it doesn't always exist in a set but if it exists this identity element then it is unique how would we go about such proof I think you can guess we can to do proof by contradiction so let's assume to the contrary so let's assume to the contrary that the identity element that the identity element is not unique it's not unique therefore under this assumption let there be at least two of them so let p1 and p2 be identity elements of some set let's make it some set p of some set p under some binary operation under some binary operation now we've just defined a few things as humans we've decided that's how we're going to define it so if my identity element is p1 I'm choosing because it's not unique I can choose any one of them and I have this binary operation with p2 that means I've got to get p2 because that's how we define identity elements and we've defined this binary this how we define this identity element before so I have to get that but if I choose p2 as my identity element then I'm going to get well that should be one if I use p so in this instance p1 now this instance of p2 is my identity element I'm going to get p1 back so in other words I have your p1 equals p1 this binary operation p2 and that's p2 in other words p2 equals p1 that is in contradiction to my initial assumption my initial assumption is therefore false therefore I have to say that the identity element if it exists it is unique so a nice proof of this uniqueness of an identity element if it exists