Let be a rational preference relation on X. Then the following hold: (a) The strict relation > is irreflexive and transitive. (b) The indifference relation is reflexive, symmetric, and transitive. (C) If x > y and yz, then x > z.

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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