The players on a football team are all weighed on a scale Th

The players on a football team are all weighed on a scale. The scale rounds the weight of every player to the nearest pound. The number of pounds read off the scale for each player is called his measured weight. The domain for each of the following relations below is the set of players on the team. For each relation, indicate whether the relation is:

reflexive, anti-reflexive, or neither

symmetric, anti-symmetric, or neither

transitive or not transitive

Exercise 6.2.2: Properties of relations dependence on the domain The players on a football team are all weighed on a scale. The scale rounds the weight of every player to the nearest pound. The number of pounds read off the scale for each player is called his measured weight. The domain for each of the following relations below is the set of players on the team. For each relation, indicate whether the relation is: reflexive, anti-reflexive, or neither symmetric, anti-symmetric, or neither transitive or not transitive (a) Player x is related to player y ifthe measured weight of player x is at least the measured weight of player y. No two players on the team have the same measured weight. (b) Player x is related to player y ifthe measured weight of player x is at least the measured weight of player y. There is at least one pair of players on the team who have the same measured weight. There is also at least one pair of players on the team who have different measured weights. (c) Player x is related to player y ifthe measured weight of player x is at least the measured weight of player y. All the players on the team have exactly the same measured weight

Solution

a.

A is related to B is A\'s weight is >= B\'s weight and every weight is unique
   * So we can say the relation is reflexive because A\'s weight is >= A\'s weight is true
   * The relation is antisymmetric because if R(a,b) with a b, then R(b,a) must not hold is true because the weight of every player in the team is different.
   * The relation is transitive since because if A\'s weight is >= B\'s weight and B\'s weight is >= C\'s weight we can clearly say A\'s weight is >= Cs weight

b.

A is related to B is A\'s weight is >= B\'s weight and every weight is unique
   * So we can say the relation is reflexive because A\'s weight is >= A\'s weight is true
   * The relation is neither symmetric nor antisymmetric because if R(a,b) holds, R(b,a) also holds is false and R(a,b) with a b, then R(b,a) must not hold is also false because the weight of every player in the team is not unique and we have a pair of player with equal weight, and another pair with different weight.
   * The relation is transitive since because if A\'s weight is >= B\'s weight and B\'s weight is >= C\'s weight we can clearly say A\'s weight is >= Cs weight

c.

A is related to B is A\'s weight is >= B\'s weight and every weight is unique
   * So we can say the relation is reflexive because A\'s weight is >= A\'s weight is true
   * The relation is symmetric nor antisymmetric because if R(a,b) holds, R(b,a) also holds because the weight of every player in the team is equal.
   * The relation is transitive since because if A\'s weight is >= B\'s weight and B\'s weight is >= C\'s weight we can clearly say A\'s weight is >= Cs weight