Problem session exercises
1. Let P = R2 = {
defined by
x
y
: x, y ∈ R} denote the plane, and consider the relation ∼ on P
p ∼ q ⇐⇒ p and q are linearly dependent.
(a) Show that ∼ is not an equivalence relation on P .
(b) Show that ∼ is an equivalence relation on P \ { 00 }. Then compute the
equivalence classes and find a complete set of equivalence class representatives.
2. Let H be a permutation group on a non-empty set S, and consider the relation ∼
of H-orbit equivalence on S defined by
x ∼ y ⇐⇒ ∃ α ∈ H : α(x) = y.
Theorem 20 shows that ∼ is an equivalence relation on S. Compute the equivalence
classes and find a complete set of equivalence class representatives for ∼ if:
(a) S = {1, 2, 3, 4}, H = S4
(b) S = {1, 2, 3, 4}, H = {(1)}
(c) S = {1, 2, 3, 4, 5}, H = {(1), (2 3), (4 5)}
(d) S = P = R2 , H = {τa : a = 2n
0 , n ∈ Z}
(e) S = P = R2 , H = {τa , τb ◦ r : a = 2n
0 ,b =
2m+1
0
, n, m ∈ Z}