Solutions Manual
Foundations of Mathematical
Economics
Michael Carter
,
c 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved
Chapter 1: Sets and Spaces
1. {1, 3, 5, 7 ... } or { ∈ : is odd }
1
1.2 Every ◻ also belongs to . ◻ also belongs to . Hence , have
∈ Every ∈
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
· 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 = {1, 2, 3, 4, 5, 6 }, = {2, 4, 6 }.
1.5 The player set is = {Jenny, Chris }. Their action spaces are
= {Rock, Scissors, Paper } = Jenny, Chris
1.6 The set of players is = 1{, 2 , .. . , .}The strategy space of
each player is the set of feasible outputs
= { ∈ ℜ + : ≤ }
where is the output of dam .
1.7 The player set is = {1, 2, 3}. There are 23 = 8 coalitions, namely
,
c 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved
( ) = {∅ , {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 ∈ ( ∪ ) . That is ∈
/ ∪ . This implies ∈/
and ∈
/ , or ∈ and ∈ . Consequently, ∈ ∩ . Conversely,
assume ∈ ∩ . This implies that ∈ and ∈
.
Consequently ∈
/ and ∈/ and therefore
◻ ∈/ ∪ . This implies that ∈ ( ∪ ) . The other identity is proved
similarly.
1.9
∪
=
∈
∩
∈ =∅
1
,
c 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved
2
1
1
-1 0 1
-1
Figure 1.1: The relation {(, ) : 2 + 2 = 1 }
1.10 The sample space of a single coin toss is {, . Th}e set of
possible outcomes in three tosses is the product
{
{, }× {, }× {, }= (, , ), (, , ), (, , ),
}
(, , ), (, , ), (, , ), (, , ), (, , )
A typical outcome is the sequence (, , ) of two heads followed by a
tail.
1.11
2
Foundations of Mathematical
Economics
Michael Carter
,
c 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved
Chapter 1: Sets and Spaces
1. {1, 3, 5, 7 ... } or { ∈ : is odd }
1
1.2 Every ◻ also belongs to . ◻ also belongs to . Hence , have
∈ Every ∈
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
· 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 = {1, 2, 3, 4, 5, 6 }, = {2, 4, 6 }.
1.5 The player set is = {Jenny, Chris }. Their action spaces are
= {Rock, Scissors, Paper } = Jenny, Chris
1.6 The set of players is = 1{, 2 , .. . , .}The strategy space of
each player is the set of feasible outputs
= { ∈ ℜ + : ≤ }
where is the output of dam .
1.7 The player set is = {1, 2, 3}. There are 23 = 8 coalitions, namely
,
c 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved
( ) = {∅ , {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 ∈ ( ∪ ) . That is ∈
/ ∪ . This implies ∈/
and ∈
/ , or ∈ and ∈ . Consequently, ∈ ∩ . Conversely,
assume ∈ ∩ . This implies that ∈ and ∈
.
Consequently ∈
/ and ∈/ and therefore
◻ ∈/ ∪ . This implies that ∈ ( ∪ ) . The other identity is proved
similarly.
1.9
∪
=
∈
∩
∈ =∅
1
,
c 2001 Michael Carter
Solutions for Foundations of Mathematical Economics All rights reserved
2
1
1
-1 0 1
-1
Figure 1.1: The relation {(, ) : 2 + 2 = 1 }
1.10 The sample space of a single coin toss is {, . Th}e set of
possible outcomes in three tosses is the product
{
{, }× {, }× {, }= (, , ), (, , ), (, , ),
}
(, , ), (, , ), (, , ), (, , ), (, , )
A typical outcome is the sequence (, , ) of two heads followed by a
tail.
1.11
2
