# ASolutions to the exercises

## A.1Chapter 1

(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:

$\frac{100}{{P}_{e}}$

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 ...

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