ECOS2903 Lecture 1: Sets
Jiemai Wu∗
Week 1
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Introduction
Goal in this class: solve general constrained optimisation problems (last 4 weeks)
• In order to get there, in the first 9 week, we will first learn an assortment of topics: open and
closed sets, functions, continuity, differentiation, integration, concavity and convexity, local
and global optima, matrix algebra...
• On its own, each of these topic is useful in economic studies, but eventually all of them will
come together as different components of one general optimisation problem.
1.1
Why math?
Why do we use math in Economics? Because the mathematical language is superior to our daily
language in economic analysis. It is more efficient and it helps us avoid logical inconsistency.
• Example: Mathematics as an efficient language.
We are all very familiar with the Demand-Supply diagram in economics:
∗
If you spot an error, please contact me at jiemai.wu@sydney.edu.au
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With this mathematical diagram, we can explain what a “market equilibrium” is in one line:
It is the intersection of the demand curve and the supply curve.
Do you know how many paragraphs Adam Smith took to explain the same idea?
Wealth of Nations by Adam Smith (1776)
Book 1, Chapter 7 Of the Natural and Market Price of Commodities
[09] When the quantity of any commodity which is brought to market falls short
of the effectual demand, all those who are willing to pay the whole value of the
rent, wages, and profit, which must be paid in order to bring it thither, cannot
be supplied with the quantity which they want. Rather than want it altogether,
some of them will be willing to give more. A competition will immediately begin among them, and the market price will rise more or less above the natural
price, according as either the greatness of the deficiency, or the wealth and wanton luxury of the competitors, happen to animate more or less the eagerness of the
competition. Among competitors of equal wealth and luxury the same deficiency
will generally occasion a more or less eager competition, according as the acquisition of the commodity happens to be of more or less importance to them. Hence
the exorbitant price of the necessaries of life during the blockade of a town or in
a famine.
[10] When the quantity brought to market exceeds the effectual demand, it cannot
be all sold to those who are willing to pay the whole value of the rent, wages,
and profit, which must be paid in order to bring it thither. Some part must be
sold to those who are willing to pay less, and the low price which they give for
it must reduce the price of the whole. The market price will sink more or less
below the natural price, according as the greatness of the excess increases more
or less the competition of the sellers, or according as it happens to be more or less
important to them to get immediately rid of the commodity. The same excess in
the importation of perishable, will occasion a much greater competition than in
that of durable commodities; in the importation of oranges, for example, than in
that of old iron.
[11] When the quantity brought to market is just sufficient to supply the effectual
demand, and no more, the market price naturally comes to be either exactly, or as
nearly as can be judged of, the same with the natural price. The whole quantity
upon hand can be disposed of for this price, and cannot be disposed of for more.
The competition of the different dealers obliges them all to accept of this price,
but does not oblige them to accept of less.
[12] The quantity of every commodity brought to market naturally suits itself to
the effectual demand. It is the interest of all those who employ their land, labour,
or stock, in bringing any commodity to market, that the quantity never should
exceed the effectual demand; and it is the interest of all other people that it never
should fall short of that demand.
• Example: Mathematical language helps us detect logical inconsistency
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Here is a logical argument - two premises and a conclusion. Try to determine as fast as you
can, if the argument is logically valid. That is, does the conclusion follow from the premises?
– All roses are flowers.
– Some flowers fade quickly.
– Therefore, some roses fade quickly.
Many of us are tempted to say “yes” because we know sometimes roses do fade quickly in
real life. But is the conclusion a correct logical outcome of the two premises?
Let’s check the answer by translating these lines into the mathematical language using a set
diagram.
Let A = all flowers, B = roses, C = flowers that fade quickly
The three lines above are translated into the following three lines:
– B ⊆ A (B is a subset of A)
– C ⊆ A (C is a subset of A)
– Therefore, B ∩C 6= ∅ (the intersection of B and C is non-empty)
Is this true?
1.2
On mathematical modelling:
In economics, we often make many specific assumptions. For example, we may assume that
our happiness is equal to the square root of our wealth, or that a firm’s production technology is
described by a linear function. Compare to the economic behaviours in real life, these assumptions
are, of course, inaccurate and over-simplified. Why do we still use them in most economic models?
Remark 1: More realistic models are better at explaining the past but worse at predicting the
future.
Compare: models of consumer demand for peaches
Model 1: demand of peaches depends on only 1 variable: price of peaches.
Model 2: demand of peaches depends on 2 variables: price of peaches and price of mangoes
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Model 3: demand of peaches depends on 3 variables: price of peaches, price of mangoes,
and your roommate’s purchase of peaches
......
Model 10: demand of peaches depends on 10 variables: price of peaches, price of mangoes,
roommate’s purchase of peaches, roommate’s purchase of other fruit, quality of peaches,
income, total expenditure on other goods, mood of the day, weather, travel time to the supermarket
Compared to the other models, Model 10 is the most realistic and can explain most of the
demand variations in the past. However, if we want to use Model 10 to predict future demand, we must collect information about all 10 variables, which is an almost impossible
task. On the other hand, Model 1 is the least accurate model, but if we want to predict the
future demand, it is the easiest model to use because we need only information on price.
Therefore, when you are tempted to criticise a model for its lack of realism, remember this:
A model that describes every detail of the current world is unable to say anything about the
future world.
Remark 2: A good economic model is not a news report but a fable: it has the perfect balance
between simplicity and realism. Economists consciously omit less relevant details of the
world to better understand its underlying principles.
For example, when an economic model that assumes
p
U(peaches, mangoes) = peaches × mangoes,
it doesn’t really mean that we believe real people should calculate square roots when they
shop at the supermarket. We make this assumption simply because the square-root function
is one of the simplest functions that can characterise peaches and mangoes as imperfect
substitutes.
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Sets
Example 1. R is the set of all real numbers.
Let n ≥ 1 be an integer. Rn is the set of all ordered n−vectors x = (x1 , x2 , ..., xn ) , where
x1 , x2 , ..., xn are all real numbers.
Notation:
x (in bold letter): a vector.
x1 , x2 , ..., xn are called the “coordinates” of x.
We also write x = (x1 , x2 , ..., xn ) as


x1
 x2 


x =  .. 
 . 
xn
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Example 2. Intervals - subsets of R
(0, 1) is an open interval.
Q1: What is the defining feature that makes it “open”?
Answer 1: This interval does not contain its end points (0 and 1).
Answer 2: Pick any number x from this open interval, you can always find numbers to the left
and right of x that still belong to (0, 1).
[0, 1] is a closed interval.
Q2: What is the defining feature that makes it “closed”?
Answer: This interval contains both end points (0 and 1).
Can we generalise these ideas of “openness” and “closedness” to any (possibly multi-dimensional)
set?
(The following definitions are from Simon and Blume Sections 12.3 - 12.4.)
Definition. ε-ball
For a vector z ∈ Rn and a positive real number ε, an ε-ball around z is the set of vectors x ∈ Rn
such that the distance between x and z is strictly less than ε:
Bε (z) = {x ∈ Rn | kx − zk < ε}
Let’s generalise Answer 2 to Q1 in the last example into the following definition for an open
set.
Definition. Open set
A set S ⊆ Rn is open if for each x ∈ S, there exists an open ε-ball around x that is completely
contained in S:
x ∈ S ⇒ there is an ε > 0 such that Bε (x) ⊆ S
Let’s generalise the answer to Q2 in the last example into the following definition for a closed
set.
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Definition. Closed set
A set is closed if it contains all of its limit points, i.e.,
a set S ⊆ Rn is closed if, whenever {xn }∞
n=1 is a convergent sequence completely contained in
S, its limit is also contained in S.
(The following definitions are from Sydsaeter, Hammond, Strome and Carvajal Section 13.5)
Definition. Interior point and boundary point
A point x is called an interior point of a set S ⊆ Rn if there exists some ε-ball around x that
is completely contained in S.
A point x is called a boundary point of a set S ⊆ Rn if every ε-ball around x contains points
of S as well as points outside of S.
Example 3. 1 is an interior point of the set (0, 2] because we can construct a small one-dimensional
ball b = (0.9, 1.1) that is centred at 1 such that all points strictly inside (0.9, 1.1) lie inside the set
(0, 2].
2 is a boundary point of the set (0, 2] because any ball centred at 2 must contain numbers bigger
than 2.
Definition. Alternative definitions for open and closed sets
A set is open if it consists only of interior points.
A set is closed if it contains all of its boundary points.
Example 4. (0, 1) is open.
[0, 1] and [0, ∞) are closed.
[0, 1) is neither open nor closed.
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The empty set ∅ is both open and closed because any statement about the empty set is trivially
true.
Proof. Let’s prove [0, ∞) is closed, and [0, 1) is neither open nor closed.
(1) [0, ∞) is closed: proof by contradiction, using the “boundary point” definition.
Suppose that [0, ∞) is not closed. This means that it has some boundary point x that is on the
outside of this interval, i.e., x ∈ (−∞, 0). Let ε = |x|. Then, the open ball ε-ball centred at x is
entirely contained in (−∞, 0). In other words, Bε (x) ∩ [0, ∞) = 0.
/ Because this open ball round x
does not contain any point in [0, ∞), x cannot be a boundary point of [0, ∞) and we have found a
contradiction.
Here is an alternative proof by contradiction, using the “limit-point” definition.
Suppose that [0, ∞) is not closed. This means that it has some limit point z that is on the outside
of this interval, i.e., z ∈ (−∞, 0). This implies that there exists some a > 0 such that
|0 − z| ≥ a
(a can be interpreted as the “gap” between z and 0.)
Because z is a limit point of [0, ∞), by the definition of limit points, there must exist some
convergent sequence {xn }∞
n=1 in [0, ∞) whose limit point is z, i.e., xn ∈ [0, ∞) for all n and
lim |xn − z| = 0.
n→∞
However, because each xn is in [0, ∞) and there is some gap a between z and 0,
|xn − z| ≥ |0 − z| ≥ a > 0 for all n.
Therefore,
lim |xn − z| ≥ a > 0
n→∞
which contradicts
lim |xn − z| = 0.
n→∞
Because we have arrived at the contradiction, our premise of “[0, ∞) is not closed” is false.
This proves that [0, ∞) is closed.
(2) [0, 1) is neither open nor closed: direct proof
“Not open”: Every ε-ball around 0 takes the form (−ε, ε). For any ε, no matter how small it
is, (−ε, ε) * [0, 1). This proves that [0,1) is not open.
“Not closed”: Here I used a boundary-point argument. The in-class notes used the limit-point
argument.
Every ε-ball around 1 takes the form (1 − ε, 1 + ε) and contains both points inside [0, 1) as
well as points outside of it. Therefore, 1 is a boundary point. Because this boundary point is not
contained in [0, 1), [0, 1) is not a closed set.
Exercise 1. R is both open and closed.
Proof. (1) R is open.
Pick any x ∈ R and any ε > 0, Bε (x) = (x − ε, x + ε) ⊆ R. This proves that R is open.
(2) R is closed.
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Pick any x ∈ R and any ε > 0, Bε (x) = (x − ε, x + ε) ⊆ R. This proves that R has no boundary
point, i.e., the set of its boundary points is 0.
/ Because 0/ is a subset of all sets, 0/ ⊆ R (the set of
boundary points of R is a subset of R) and this proves that R is closed.
Alternatively, R is closed because any limit of a convergent real-number sequence must also
be a real number, so R contains all of its limit points.
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