Foundations of Mathematical Economics By Michael Carter

Solutions Manual Foundations of Mathematical
Economics By Michael Carter

Descriptive method of Set Representation, example: the set of all integers between 8
and 15 (both inclusive), the set of all real numbers between 8 and 15 (both inclusive)
- ANSWER:{x| x≤8≤15, x∈ZZ}
{x| 8≤x≤15, x∈R}

Enumeration method of Set Representation, example: the set of all integers between
8 and 15 (both inclusive), the set of all real numbers between 8 and 15 (both
inclusive) - ANSWER:{8, 9, ..., 14, 15}
Not possible, |R| = x1, the reals are infinitely uncountable and therfore are not
enumerable

Standard method of Set Representation, example: the set of all integers between 8
and 15 (both inclusive), the set of all real numbers between 8 and 15 (both inclusive)
- ANSWER:The set A = [8,15]

aleph naught - ANSWER:X_0, infinitely countable, example |N|, enumerable

cardinality of a set - ANSWER:size of a set, |A|

aleph_1 - ANSWER:X_1 infinitely uncountable, not emumerable, |R|

Union Set - ANSWER:Given set A and B, A ∪ B = C = {x| x∈A → x∈C or x∈B → x∈C,
or x∈A ∧ x∈B → x∈C} = the set of all elements in A, or B or A and B

Intersection Set - ANSWER:Given set A and B, A ∩ B = C = {x| for all x∈A ∧ x∈B →
x∈C} = the set of all elements in A and B.

Difference Set - ANSWER:Given set A and B, A\B = C = {x| for all x∈A ∧ x ∉B → x∈C}
= the set of all elements in A but not in B.

Universal Set - ANSWER:U, set of everything

Set of Integers - ANSWER:ZZ = all whole numbers and their negative counter parts,
{0, -1, 1, -2, 2, -3, 3, ...}

What is a group - ANSWER:group of distinct objects
A = {1,2,3}
B = {Orange, Banana, Apple}
R

Set of Real Numbers - ANSWER:ℝ, set of rational numbers ∪ set of irrational
numbers

, Set of Natural Numbers - ANSWER:ℕ, {1, 2, 3, ...}

Set of Rational Numbers - ANSWER:ℚ, {x| x = a/b, a∈ZZ ∧ b∈ZZ ∧ b ≠ 0}

Set of Complex Numbers - ANSWER:ℂ, {x| x=a+bi, a∈R, b∈R, i²=-1}

Null Set - ANSWER:empty set, ∅

a number is even if - ANSWER:x is even if ∃k ∈ ZZ | x = 2k

a number is odd if - ANSWER:x is even if ∃k ∈ ZZ | x = 2k + 1

Set of Irrational numbers - ANSWER:{x| !∃a∈ZZ, !∃b∈ZZ | x=a/b, b≠0 or ∀a∈ZZ,
∀b∈ZZ, | x≠a/b, b≠0}

a number is rational if - ANSWER:x is rational if ∃a∈ZZ, ∃b∈ZZ | x=a/b, b≠0

a number is irrational if - ANSWER:x is irrational if ¬∃a∈ZZ, ¬∃b∈ZZ | x=a/b, b≠0 or
∀a∈ZZ, ∀b∈ZZ, | x≠a/b, b≠0

Whole Numbers - ANSWER:The set of the natural numbers and zero

a number is prime if - ANSWER:a number x ∈ ZZ+ is prime if the only positive factors
of p are p and 1

a number is composite if - ANSWER:a number x is not prime

Set T⊆S - ANSWER:T⊆S → ∀x∈T →x∈S, every element of T is an element of S, S
contains T

Set T=S - ANSWER:T=S → T⊆S ∧ S⊆T

Set T¬⊆S - ANSWER:T¬⊆S → ∃x∈T | x∉S

Set T ⊂ S - ANSWER:T⊆S and S≠T

Powerset - ANSWER:P(S) = set of all subsets of S
|P(S)| = 2^|S|

complement set - ANSWER:A^c = {x| x∈U ∧ x∉A → x ∈ A^c}

Set Domination Laws, A ∪ U = ?, A ∩ ∅ = ? - ANSWER:A ∪ U = U, A ∩ ∅ = ∅

Set Identity Laws, A ∩ U = ?, A ∪ ∅ = ? - ANSWER:A ∩ U = A, A ∪ ∅ = A ?

Set Idenpotent Laws, A ∪ A = ?, A ∩ A = ? - ANSWER:A ∪ A = A, A ∩ A = A