Page 26
R
x
y
y
x
x
x!y
y
x
x
z
x
y
z
y
y
Page 27
Page 28
x
x!y
x
Z+
y
Z
y
Page 29
R
R∗
A
A
R∗
R ⊆ R∗
S
A
R⊆S
R∗ ⊆ S
R
Page 30
A = {1, 2, 3}
R = {(1, 1), (1, 2), (1, 3), (2, 3), (3, 1)}.
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1
2
3
5
4
6
Page 32
R
A
R
xRy
Ex
Ex = {y | yRx}.
x∈A
xy > 0
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Page 34
R
R
xRy
x−y
E0 = Z
R
0
E 1 = {. . . , −2 12 − 1 12 , − 12 , 12 , 1 12 , 2 12 , . . .}
2
1
2
Page 35
A
A1 , . . . , An
A
A = A1 ∪ A2 ∪ · · · ∪ An
Ai ∩ Aj = ∅
i!j
Ai
A
A1
A2
A4
A3
A
Page 36