⃝c 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved
1
, ⃝c 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved
1
, ⃝c 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved
1
, ⃝c 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved
Chapter 1: Sets and Spaces
1.1
{ 1, 3, 5, 7 . . . } or { 𝑛 ∈ 𝑁 : 𝑛 is odd }
1.2 Every 𝑥 ∈ 𝐴 also belongs to 𝐵. Every 𝑥 ∈ 𝐵 also belongs to 𝐴. Hence 𝐴, 𝐵 have
precisely the same elements.
1.3 Examples of finite sets are
∙ the letters of the alphabet { A, B, C, . . . , Z }
∙ the set of consumers in an economy
∙ the set of goods in an economy
∙ the set of players in a game.
Examples of infinite sets are
∙ the real numbers ℜ
∙ the natural numbers 𝔑
∙ the set of all possible colors
∙ the set of possible prices of copper on the world market
∙ the set of possible temperatures of liquid water.
1.4 𝑆 = { 1, 2, 3, 4, 5, 6 }, 𝐸 = { 2, 4, 6 }.
1.5 The player set is 𝑁 = { Jenny, Chris }. Their action spaces are
𝐴𝑖 = { Rock, Scissors, Paper } 𝑖 = Jenny, Chris
1.6 The set of players is 𝑁 = {1, 2 , . . . , 𝑛 .} The strategy space of each player is the set
of feasible outputs
𝐴𝑖 = { 𝑞 𝑖 ∈ ℜ + : 𝑞 𝑖 ≤ 𝑄 𝑖 }
where 𝑞𝑖 is the output of dam 𝑖.
1.7 The player set is 𝑁 = {1, 2, 3}. There are 23 = 8 coalitions, namely
𝒫(𝑁) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
There are 210 coalitions in a ten player game.
1.8 Assume that 𝑥 ∈ (𝑆 ∪ 𝑇 )𝑐. That is 𝑥 ∈ / 𝑆 ∪ 𝑇 . This implies 𝑥 ∈ / 𝑆 and 𝑥 ∈ / 𝑇,
or 𝑥 ∈ 𝑆𝑐 and 𝑥 ∈ 𝑇 𝑐. Consequently, 𝑥 ∈ 𝑆𝑐 ∩ 𝑇 𝑐. Conversely, assume 𝑥 ∈ 𝑆𝑐 ∩ 𝑇 𝑐.
This implies that 𝑥 ∈ 𝑆𝑐 and 𝑥 ∈ 𝑇 𝑐. Consequently 𝑥 ∈ / 𝑆 and 𝑥 ∈ / 𝑇 and therefore
𝑥∈/ 𝑆 ∪ 𝑇 . This implies that 𝑥 ∈ (𝑆 ∪ 𝑇 ) 𝑐 . The other identity is proved similarly.
1.9
∪
𝑆=𝑁
𝑆∈𝒞
∩
𝑆=∅
𝑆∈𝒞
1
Solutions for Foundations of Mathematical Economics All rights reserved
1
, ⃝c 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved
1
, ⃝c 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved
1
, ⃝c 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved
Chapter 1: Sets and Spaces
1.1
{ 1, 3, 5, 7 . . . } or { 𝑛 ∈ 𝑁 : 𝑛 is odd }
1.2 Every 𝑥 ∈ 𝐴 also belongs to 𝐵. Every 𝑥 ∈ 𝐵 also belongs to 𝐴. Hence 𝐴, 𝐵 have
precisely the same elements.
1.3 Examples of finite sets are
∙ the letters of the alphabet { A, B, C, . . . , Z }
∙ the set of consumers in an economy
∙ the set of goods in an economy
∙ the set of players in a game.
Examples of infinite sets are
∙ the real numbers ℜ
∙ the natural numbers 𝔑
∙ the set of all possible colors
∙ the set of possible prices of copper on the world market
∙ the set of possible temperatures of liquid water.
1.4 𝑆 = { 1, 2, 3, 4, 5, 6 }, 𝐸 = { 2, 4, 6 }.
1.5 The player set is 𝑁 = { Jenny, Chris }. Their action spaces are
𝐴𝑖 = { Rock, Scissors, Paper } 𝑖 = Jenny, Chris
1.6 The set of players is 𝑁 = {1, 2 , . . . , 𝑛 .} The strategy space of each player is the set
of feasible outputs
𝐴𝑖 = { 𝑞 𝑖 ∈ ℜ + : 𝑞 𝑖 ≤ 𝑄 𝑖 }
where 𝑞𝑖 is the output of dam 𝑖.
1.7 The player set is 𝑁 = {1, 2, 3}. There are 23 = 8 coalitions, namely
𝒫(𝑁) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
There are 210 coalitions in a ten player game.
1.8 Assume that 𝑥 ∈ (𝑆 ∪ 𝑇 )𝑐. That is 𝑥 ∈ / 𝑆 ∪ 𝑇 . This implies 𝑥 ∈ / 𝑆 and 𝑥 ∈ / 𝑇,
or 𝑥 ∈ 𝑆𝑐 and 𝑥 ∈ 𝑇 𝑐. Consequently, 𝑥 ∈ 𝑆𝑐 ∩ 𝑇 𝑐. Conversely, assume 𝑥 ∈ 𝑆𝑐 ∩ 𝑇 𝑐.
This implies that 𝑥 ∈ 𝑆𝑐 and 𝑥 ∈ 𝑇 𝑐. Consequently 𝑥 ∈ / 𝑆 and 𝑥 ∈ / 𝑇 and therefore
𝑥∈/ 𝑆 ∪ 𝑇 . This implies that 𝑥 ∈ (𝑆 ∪ 𝑇 ) 𝑐 . The other identity is proved similarly.
1.9
∪
𝑆=𝑁
𝑆∈𝒞
∩
𝑆=∅
𝑆∈𝒞
1
