1. (20 points) Consider a group of individuals A, B and C and the relation “as wealthy as” like A is as wealthy as B.
(a) Does this relation satisfy the completeness and transitivity properties?
(b) Take the same group of individuals as above and consider the relation strictly wealthier than. Is this relation transitive?
(a) The relation "as wealthy as" does not satisfy the completeness property. This is because there is no comparison made between A and C in terms of wealth. The relation only states that A is as wealthy as B, but it does not provide information about how wealthy A is compared to C.
The relation "as wealthy as" satisfies the transitivity property. If A is as wealthy as B, and B is as wealthy as C, then it follows that A is as wealthy as C. The transitive property holds for this relation.
(b) If we consider the relation "strictly wealthier than" in the same group of individuals, it does not necessarily satisfy the transitivity property. For example, if A is strictly wealthier than B, and B is strictly wealthier than C, it does not imply that A is strictly wealthier than C.
Transitivity requires that if A > B and B > C, then A > C for all elements A, B, and C in the relation. However, in this case, it is possible for A to be wealthier than B, and B to be wealthier than C, but A may not be strictly wealthier than C. Transitivity does not hold for this relation.