slakedlime
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Homework Statement
This is not a homework problem, but a topic in a microeconomics book that I am unclear about.
My book argues that the set X = {a, b, c, d} of preferences can be (i) transitive but (ii) incomplete.
Is it possible for a similar set of preferences to be (i) complete but (ii) intransitive, depending on how we define the relations between the elements? My book does not go into this, but I am curious, and any explanation would be much appreciated.
Homework Equations
(i) Completeness axiom: For every pair x, y [itex]\in[/itex] X, either x is weakly preferred to y, y is weakly preferred to x, or both (that is, we are indifferent between x and y).
(ii) Transitivity axiom: For every triple x, y, z [itex]\in[/itex] X, if x is weakly preferred to y and y is weakly preferred to z, then x is weakly preferred to z.
The Attempt at a Solution
I understand that the set X = {a, b, c, d} can be transitive but incomplete. My book explains this with a table (picture attached; my book says that this is only one way of proving that the set X can be transitive but incomplete, and that there are many other ways); the incompleteness arises because we have not defined a relationship between c and d, making them incomparable.
However, the set is transitive because (i) a is weakly preferred to b, and b is weakly preferred to c; (ii) a is weakly preferred to b, and b is weakly preferred to d; (iii) a is weakly preferred to c and a is also weakly preferred to d.
I don't understand how it is possible for a set to be complete but intransitive. If we have defined relationships between all the elements such that any two pairs are comparable, doesn't transitivity automatically follow? Is my understanding correct?
