Quasitransitivity is defined in the problem itself. We say that the preference relation (R) is Quasitransitive if its associated strict preference relation (P) is transitive i.e.for all x,y,z belonging to X xPy and yPz implies xPz.
First let me tell you from the list which ones are not quasitransitive and thats not very hard to get: The strict preference relation(P) does not satisfy tansitivity in the following:
a) xPy & yPz & zPx (because xPy yPz implies xPz which is not the case here)
b) xPy & yPz & zIx (because xPy yPz implies xPz which is not the case here)
d) yPx & yIz & xPz (because yPx xPz implies yPz which is not the case here)
By elimination its c).
To deduce that its c) directly is also simple. As we can clearly see that quasitransitivity is not violated here: If R is such that the associated P and I respects xPy & yIz & zIx then there is just one pair (x,y) in relation P so there is no way we can say that P is not transitive.
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