This document defines a set function (N, v) that assigns values to subsets of a set N. It proves several properties of such set functions, including: 1) The value of the entire set N is greater than the sum of individual element values 2) Translating a set function (N, v) to a new function (N, v') by scaling and shifting preserves properties like submodularity 3) A vector x dominates another y with respect to a set S if each element of x is greater than the corresponding element of y for all elements in S. 4) Translating vectors preserves dominance relationships defined by the original set function.

1. 2

2. 1.3

3. 1.3
(a)
(b)
(c)0-

4. 3

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)

10.
(N, v) (N, v′)
v′(S) = αv(S) +
∑
i∈S
βi, α > 0
(N, v) x = (x1, x2, ⋯, xn)
x′i = αxi + βi, i = 1,2,⋯, n
(N, v′) x′ = (x′1, x′2, ⋯, x′n)
ϕ : x → x′ (N, v) (N, v′)

11.
ϕ : x → x′ (N, v) (N, v′)
(N, v) 𝒜 (N, v′) 𝒜′
ϕ(x) = αx + β
a, b ∈ 𝒜 ϕ(a) = ϕ(b)
αa + β = αb + β ⇔ a = b
a, b ∈ 𝒜 a ≠ b ⇒ ϕ(a) ≠ ϕ(b)
b ∈ 𝒜′ a =
1
α
b −
β
α
ϕ(a) = αa + β = α(
1
α
b −
β
α
) + β = b
ϕ(𝒜) = 𝒜′
𝒜′ 𝒜

12.
(N, v) (N, v′) x y (N, v)
x′ y′ (N, v′) S xdomSy
x′domSy′
xdomSy v′(S) ≥
∑
i∈S
x′(i) x′i > y′i, ∀i ∈ S
xdomSy v(S) ≥
∑
i∈S
x(i) ⇔
v′(S) − ∑i∈S
βi
α
≥
∑
i∈S
x′− βi
α
⇔ v′(S) −
∑
i∈S
βi ≥
∑
i∈S
(x′− βi) ⇔ v′(S) ≥
∑
i∈S
x′(i)
xi > yi, ∀i ∈ S ⇔
x′i − βi
α
>
y′i − βi
α
, ∀i ∈ S
⇔ x′i > y′i, ∀i ∈ S
x′domSy′
(N, v) x, y S
v(S) ≥
∑
i∈S
x(i)
xi > yi, ∀i ∈ S
x S y x domS y
x′i = αxi + βi, i = 1,2,⋯, n

13.
(N, v) (N, v′) C, C′
x′i = αxi + βi, i = 1,2,⋯, n
C, C′
(N, v) x C x′ (N, v′)
C′
(N, v) (N, v′)
C, C′

14.
(N, v)
v({1,2}) + v({2,3}) + v({1,3}) ≤ 2v({1,2,3})
x
v({1,2,3}) = x1 + x2 + x3
≤ v({1,2,3}) − v({2,3}) + v({1,2,3}) − v({1,3}) + v({1,2,3}) − v({1,2})
⇔ v({1,2}) + v({2,3}) + v({1,3}) ≤ 2v({1,2,3})
x1 + x2 + x3 = v({1,2,3})
x1 + x2 ≥ v({1,2})
x2 + x3 ≥ v({2,3})
x1 + x3 ≥ v({1,3})
x1, x2, x3 ≥ 0
x1 + x2 + x3 = v({1,2,3})
0 ≤ x1 ≤ v({1,2,3}) − v({2,3})
0 ≤ x2 ≤ v({1,2,3}) − v({1,3})
0 ≤ x3 ≤ v({1,2,3}) − v({1,2})

15.
(N, v)
v({1,2}) + v({2,3}) + v({1,3}) ≤ 2v({1,2,3})
v({1,2}) + v({2,3}) + v({1,3}) ≤ 2v({1,2,3})
v({1,3}) ≥ v({1,2,3})
v({1,2}) + v({2,3}) + v({1,3}) ≤ 2v({1,2,3}) ≤ v({1,2,3}) + v({1,3}) ⇔ v({1,2}) + v({2,3}) ≤ v({1,2,3})
x1 = v({1,2,3}) − v({2,3}), x2 = 0, x3 = v({2,3})
v({1,2}) + v({2,3}) > v({1,2,3})
x1 = v({1,2,3}) − v({2,3}), x2 = v({1,2}) + v({2,3}) − v({1,2,3}), x3 = v({1,2,3}) − v({1,2})
x1 + x2 + x3 = v({1,2,3})
0 ≤ x1 ≤ v({1,2,3}) − v({2,3})
0 ≤ x2 ≤ v({1,2,3}) − v({1,3})
0 ≤ x3 ≤ v({1,2,3}) − v({1,2})
v({2,3}) ≤ v({1,2,3}) − v({1,2})
v({1,2}) + v({2,3}) − v({1,2,3}) ≤ v({1,2,3}) − v({1,3})

17. See you next time
2