 So we continue our discussion on this ordering of sets and I've got a question for you in a statement So the first question is one that when does a relation partially order a set and the total ordering is not included It's when the following holds when the following holds that R is reflexive that relation is reflexive reflexive Spelling seems to be a problem today. So R is reflexive when R is anti-symmetric Anti-symmetric that's a new one. What does that anti-symmetric mean? Well, if we have a sub a sub one and it's relation two and we have two and one That can occur if and only if if and only if a one equals a two So there's this constraint on this symmetric property So only when these two are equal do we actually have that and if they are equal obviously we set with a reflexive property And then the relation must also be transitive Transitive so transitive spelling really a problem today anyway long long long night So it must have these properties for that to partially order a set and then of course total ordering is not included Because look at the statement if a is a partially ordered set with respect to some relation Then every subset of a is at least partially ordered with respect to that same relation so remember we had the set and the set was one two three four twelve and our relation was as divides So it divides and we could write out and that there's a nice way to do this one two for 12 and that three and three to twelve That we do have that partial ordering because let me take one subset a sub one and if a sub one is one two three Well, what do I have I have one? one divides two and One divides so for the noise one divides three there Two certainly does not divide three. So we have partial ordering here. Remember we have We have a first element. We don't have a last element first element is unique We have a minimal which is one and we have two maximals which are two and three But I can have the subset. Let's make the subset a two and that subset might be one two and four And certainly I have one two and four one divide itself one divides two one divides four two divides four and two And four divides four In other words, this will be That will be totally ordered versus this partial ordering So that's why I say at least partial ordered in any of the subsets that you can take there on the same relation If the original one is partially ordered all its subset will be at least partially ordered