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1. 1

2. 1.
2.

3. 1.
(a)
(b)
(c)
(d)0- (0, 1)-
(e)
2.

5.
N
ν N
ν (N, v)
N = {1, 2, 3}
ν({1,2,3}) = 100
ν({1,2}) = 50

6.
(N, ν)
S ⊃ T
v
v
v
v
v(S ∪ T) ≥ v(S) + v(T)
v(S) ≥ v(T)
v(N) >
n
∑
i=1
v(i)
v(N) = v(S) + v(NS)

7.
(N, ν) (N, ν′)
α β1, ⋯, βn S
ν′(S) = αν(S) +
∑
iz∈S
βi

8.
(N, ν)
ν′(i) = 0, i = 1,⋯ . n (1)
ν′(N) = 1, ν′(i) = 0, i = 1,⋯, n (2)
(N, ν′)
(N, ν′)
(N, ν′)

9.
x = (x1, ⋯, xn)
xi ≥ v(i) ∀i ∈ N
∑
i∈N
xi = v(N)

11.
(N, v) x, y S
v(S) ≥
∑
i∈S
x(i)
xi > yi, ∀i ∈ S
x S y x domS y
(N, v)
S
{xi}i∈S
S
{xi}i∈S {yi}i∈S

12.
(N, ν)
∑
i∈S
x(i) ≥ v(S), ∀S ⊂ N

13.
(N, ν)
∑
i∈S
x(i) ≥ v(S), ∀S ⊂ N ( * )
( * ) x x y
S y domS x
v(S) ≥
∑
i∈S
y(i) >
∑
i∈S
x(i)
( * ) ( * ) x
v(S) ≥
∑
i∈S
y(i)
yi > xi, ∀i ∈ S

14.
(N, ν)
∑
i∈S
x(i) ≥ v(S), ∀S ⊂ N ( * )
x S ( * )
ϵ = v(S) −
∑
i∈S
xi > 0, α = v(N) − (v(S) +
∑
j∈NS
v(j))
α ≥ 0
j1, j2 ∈ NS
v(j1 ∪ j2) ≥ v(j1) + v(j2)
v(j1 ∪ j2 ∪ j3) ≥ v(j1 ∪ j2) + v(j3) ≥ v(j1) + v(j2) + v(j3)
v(j1 ∪ j2 ∪ ⋯ ∪ j|NS|) ≥ v(j1) + v(j2)⋯ + v(j|NS|) =
∑
j∈NS
v(j)) ⇔ v(NS) ≥
∑
j∈NS
v(j))
v(S) + v(NS) ≥ v(S) +
∑
j∈NS
v(j) ⇒ v(N) ≥ v(S) + v(NS) ≥ v(S) +
∑
j∈NS
v(j) ⇒ v(N) − v(S) −
∑
j∈NS
v(j) ≥ 0
v(S ∪ T) ≥ v(S) + v(T)

15.
ϵ = v(S) −
∑
i∈S
xi > 0, α = v(N) − (v(S) +
∑
j∈NS
v(j))
y = (y1, ⋯, yn)
yi = xi +
ϵ
|S|
, if i ∈ S
= v(i) +
α
(n − |S|)
, if i ∈ NS
∑
i∈N
yi =
∑
i∈S
yi +
∑
j∈NS
yj =
∑
i∈S
(xi +
ϵ
|S|
) +
∑
j∈NS
(v(j) +
α
(n − |S|)
)
=
∑
i∈S
xi +
∑
i∈S
ϵ
|S|
+
∑
j∈NS
v(j) +
∑
j∈NS
α
(n − |S|)
=
∑
i∈S
xi +
ϵ
|S|
|S| +
∑
j∈NS
v(j) +
α
(n − |S|)
(n − |S|) =
∑
i∈S
xi + ϵ +
∑
j∈NS
v(j) + α =
∑
i∈S
xi + v(S) −
∑
i∈S
xi +
∑
j∈NS
v(j) + α
= v(S) +
∑
j∈NS
v(j) + v(N) − v(S) −
∑
j∈NS
v(j) = v(N)

16.
ϵ = v(S) −
∑
i∈S
xi > 0, α = v(N) − (v(S) +
∑
j∈NS
v(j))
y = (y1, ⋯, yn)
yi = xi +
ϵ
|S|
, if i ∈ S
= v(i) +
α
(n − |S|)
, if i ∈ NS
i ∈ S yi = xi +
ϵ
|S|
≥ v(i) +
ϵ
|S|
> v(i)
i ∈ NS yi = v(i) +
α
(n − |S|)
≥ v(i)
α ≥ 0
ϵ > 0

17.
(N, ν)
∑
i∈S
x(i) ≥ v(S), ∀S ⊂ N ( * )
x S ( * )
ϵ = v(S) −
∑
i∈S
xi > 0, α = v(N) − (v(S) +
∑
j∈NS
v(j))
α ≥ 0
y = (y1, ⋯, yn)
yi = xi +
ϵ
|S|
, if i ∈ S
= v(i) +
α
(n − |S|)
, if i ∈ NS
i ∈ S yi > xi
∑
i∈S
yi =
∑
i∈S
(xi +
ϵ
|S|
) =
∑
i∈S
xi +
∑
i∈S
ϵ
|S|
=
∑
i∈S
xi + ϵ =
∑
i∈S
xi + v(S) −
∑
i∈S
xi = v(S)
y domS x x

19. See you next time
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