Ans. a. The preference relation ≻ is irreflexive and transitive.
Let there are two commodity bundles x and y
X ≻ Y implies that the relation ≻ is transitive. Similarly if Y ≻ X then also it is transitive.
Now, X≻ X is anti-reflexive. X cannot be preferred to X. X is thus irreflexive. Same for Y≻ Y.
Ans b. ∼ is reflexive, transitive, and symmetric.
X∼X is reflexive, that is X is indifferent to X, which is true since it is the same commodity.
X∼Y implies X and Y are indifferent which also holds.
X∼Y and Y∼X imply the same thing, so it is symmetric.
Ans c. X ≻Y implies X is preferred to Y strictly.
Y ≻∼Z implies that Y is preferred to Z but not strictly i.e sometimes they are indifferent.
which obviously implies that X ≻ Z in terms of ranking the bundles i.e X>Y>=Z