Mathematical Methods of Economic Analysis
Lecture Week 1
Jingni Yang
University of Sydney
Instructor
Name: Jingni Yang
Office: Room 550, Social Science Building A2
Email: jingni.yang@sydney.edu.au
Office hour: Wednesday 1-2pm in person or by appointment
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Assessment
Two homework assignments, A mid-semester test and final exam
Weighting:
Two homework assignments: 20 %
Mid-semester test: 30 %
Final Exam: 50 %
Timeline:
Two homework assignments: week 3 and week 10 due in the next.
Mid-semester test: 1 hour online test completed during class time week 7.
Final exam: 2 hours during exam period.
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Questions
Please don’t use email for instructional purposes
Post your questions on Ed Discussion (Canvas)
Attendance at lectures is recommended but not mandatory
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Reading Material
Primary reference
Lecture slides
Readings other than lectures slides all supplementary
Mathematics for Economists by Simon and Blume
Google for some notes/textbook/exercises
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Definitions and Facts
The lectures are full of definitions and facts
For example: Function f : R→ R is called continuous at x if, for any sequence
{xn} converging to x we have f (xn)→ f (x)
Possible exam question: Show that if functions f and g are continuous at x , then
so is f + g .
Start answer with the definition of continuity:
“Let {xn} be any sequence converging to x . We need to show that
f (xn) + g(xn)→ f (x) + g(x). To see this, note that...”
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Facts: In the lectures you’ll often see something like
The only N-dimensional subset of RN is RN
Means any one of
theorem
proposition
lemma
true statement
All well-know results
Need to remember, have some intuition for, be able to apply
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Logical Operations
Logic are maily four words
NOT (or Negation), AND, OR, IF....THEN
Some of these have a somewhat different meaning in logic relative to their
meaning in ordinary spoken language.
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AND
Tom and Jerry were at movies.
I would like fish and chips.
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OR
For the main course, would you like fish or meat?
ECON5001 or ECON6001 is a pre-requisite CON6003
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IF, THEN
(S) If the GPS current location is 33S 151E, then you are in Sydney
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Equivalent forms of P =⇒ Q
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Equivalent forms of P =⇒ Q is not true
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True Values of an implication (”IF, THEN” statement)
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Negation
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Different Types of ” Proofs”
The concept of a ”proof” is a key element in math.
Easy to sliop into philosohpical debates on ” what exactly is a proof”.
A proof is a logical consequence of accepted facts (propositions).
Four types of proofs
1 Constructive/deductive proof
2 Contraposition
3 Induction
4 Contradiction
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Proof by Induction
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Proof by Contradiction I
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Proof by Contradiction II
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Proof by Contradiction III
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Sets
Will often refer to the real numbers, R
Understand it to contain “all of the numbers” on the “real line”
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Sets
R is an example of a set
A set is a well-specified collection of distinct objects viewed as a whole
(In case of R, the objects are numbers)
Other examples of sets:
set of all rectangles in the plane
set of all prime numbers
set of monkeys in Japan
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Notation:
Sets: A,B,C
Elements: x , y , z
Important sets:
N := {1, 2, 3, . . .}
Z := {. . . ,−2,−1, 0, 1, 2, . . .}
Q := {p/q : p, q ∈ Z, q 6= 0}
R := Q ∪ { irrationals }
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Intervals of R
Common notation:
(a, b) := {x ∈ R : a < x < b}
(a, b] := {x ∈ R : a < x ≤ b}
[a, b) := {x ∈ R : a ≤ x < b}
[a, b] := {x ∈ R : a ≤ x ≤ b}
[a,∞) := {x ∈ R : a ≤ x}
(−∞, b) := {x ∈ R : x < b}
Etc.
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Let A and B be sets Statement x ∈ A means that x is an element of A
A ⊂ B means that any element of A is also an element of B
N ⊂ Z
irrationals are a subset of R
A = B means that A and B contain the same elements
Equivalently, A ⊂ B and B ⊂ A
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Set Operations
Let S be a set and A and B be subsets of S
Union of A and B
A ∪ B := {x ∈ S : x ∈ A or x ∈ B}
Intersection of A and B
A ∩ B := {x ∈ S : x ∈ A and x ∈ B}
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Set Operations
Set theoretic difference:
A \ B := {x ∈ S : x ∈ A and x /∈ B}
In other words, all points in A that are not points in B
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Set Operations
Complement of A is all elements of S that are not in A:
Ac := S \ A :=: {x ∈ S : x /∈ A}
Remarks:
Need to know what S is before we can determine Ac
If not clear better write S \ A
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Algebra of Sets
If A and B subsets of S , then
1 A ∪ B = B ∪ A and A ∩ B = B ∩ A
2 (A ∪ B)c = Bc ∩ Ac and (A ∩ B)c = Bc ∪ Ac
3 A \ B = A ∩ Bc
4 (Ac)c = A
The empty set ∅ is the set containing no elements
If A ∩ B = ∅, then A and B said to be disjoint
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Infinite Unions and Intersections
Given a family of sets Kλ ⊂ S with λ ∈ Λ,⋂
λ∈Λ
Kλ := {x ∈ S : x ∈ Kλ for all λ∈Λ⋃
λ∈Λ
Kλ := {x ∈ S : there exists an λ ∈ Λ such that x ∈ Kλ}
“there exists” means “there exists at least one”
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Example I
Let A := ∩n∈N(0, 1/n)
Claim: A = ∅
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Example II
For any a < b we have ∪>0 [a + , b) = (a, b)
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