Super helpful, what a great series. Can you kindly clarify why the Total order, relations are automatically reflexive. I couldn't quite understand your comment, and was trying to prove It from the stated qualities. (I also couldn't find anteing online) Thanks!!!
This follows from just the fact that a total order is complete -- i.e., for any x and y either xRy or yRx (or both). In particular, the x and y could be the same element -- the definition doesn't refer only to *distinct* elements -- in which case we get either xRx or xRx, i.e., definitely xRx, and this is true for every x.
Also I'm confused by the fact for example 3 and 4 you are using the same notation, the curvy ≿, but defining them differently. E.g. 3 is antisymmetric but 4 isn't. Surely the curvy ≿ should be for 4, and 3 should just be ≥ ? I know that for vectors x,y x≥y implies x≿y. But surely when you come to do the upper contour for case 3, which is two boxes, the notation ≥ makes more sense. As ≿ suggests preferences, and hence the indifference curve previously drawn? thanks again!
This was just a choice of notation, and x≿y would have been fine instead, perhaps clearer, since the next example used the curly symbol for something different. Thanks for following along closely enough to have questions.
Is a total order also reflexive? Per wikipedia and NIST a total order is also reflexive. Also, if a total order is a type of partial order (per lecture 42B) then, again, this implies that it must be reflexive.
