ASolutions to the exercises
(1.1)The relation is an equivalence relation:
–Reflexivity: x attends the same class as x.
–Symmetry: If x attends the same class as y, then y also attends the same class as x.
–Transitivity: If x attends the same class as y and y attends the same class as z, then x and z also attend the same class.
(1.2)The relation is not transitive and hence not an equivalence relation, as seen in this family tree:
Alice and Bob share a grandparent (Joel), and Bob and Carol share a grandparent (Kathie), but Alice and Carol do not.
(1.3)The relation x ∼ y ⇔ xT y = 0 is not reflexive for x ≠ 0 and hence not an
(1.4)x ∼ y ⇔ xT y ≥ 0 is reflexive and symmetric, but not transitive and hence not an equivalence ...