Economics DSC 151 Unit 1 | 2nd Semester Notes

1. Set and set operations
2. Venn diagram
3. Cartesian product
4. Relations
5. Functions and their properties
6. Basic logarithmic
7. Limit of a function
8. Continuity

1. Set and set operations

A set is a well-defined collection of distinct objects. The objects in a set are called its elements or members.

Methods of Describing a Set:

Roster or Tabular Method: Listing all the elements, separated by commas and enclosed in curly braces {}.
Example: A = {1, 2, 3, 4, 5}
Set-Builder or Rule Method: Describing the set by a property that its elements must satisfy.
Example: B = {x | x is an even number and x < 10} (Read as: "x such that x is an even number and x is less than 10")

Important Types of Sets:

Empty Set (or Null Set): A set with no elements, denoted by {} or ∅.
Subset (⊆): Set A is a subset of set B (A ⊆ B) if every element of A is also an element of B.
Proper Subset (⊂): A is a proper subset of B (A ⊂ B) if A ⊆ B and A ≠ B.
Universal Set (U): The set containing all elements under consideration in a particular problem.

Set Operations:

Operation	Symbol	Definition	Example (A={1,2,3}, B={3,4,5})
Union	A ∪ B	All elements that are in A, or in B, or in both.	{1, 2, 3, 4, 5}
Intersection	A ∩ B	All elements that are in both A and B.	{3}
Difference	A - B	All elements that are in A but not in B.	{1, 2}
Complement	A' or A^c	All elements in the Universal Set (U) that are not in A. (If U={1,2,3,4,5,6}, A' = {4,5,6})	{4, 5, 6} (assuming U={1..6})

De Morgan's Laws: These are frequently asked in exams!

(A ∪ B)' = A' ∩ B'
(A ∩ B)' = A' ∪ B'

2. Venn diagram

A Venn diagram is a visual representation of sets, showing the relationships between them. The Universal Set (U) is typically represented by a rectangle, and subsets are represented by circles inside it.

[Diagram: Venn Diagram Basics]
A rectangle labeled 'U' containing two overlapping circles labeled 'A' and 'B'.

The overlapping region is A ∩ B.
The entire area of both circles combined is A ∪ B.
The part of circle A not overlapping is A - B.
The area outside both circles but inside U is (A ∪ B)'.

Economic Application:

Let A = {Households with a car}
Let B = {Households with a house}
A ∩ B = {Households with both a car and a house}
A ∪ B = {Households with at least one of the assets}

3. Cartesian product

Definition: The Cartesian product of two non-empty sets A and B, denoted A × B, is the set of all possible ordered pairs (a, b) where a ∈ A and b ∈ B.
A × B = {(a, b) | a ∈ A and b ∈ B}

Example:
Let A = {1, 2} and B = {a, b, c}
Then, A × B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}

Example:
Let B = {a, b, c} and A = {1, 2}
Then, B × A = {(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2)}

Order Matters! The Cartesian product is not commutative.
As seen in the example, A × B ≠ B × A (unless A = B or one is ∅).

If set A has m elements and set B has n elements, then A × B has m * n elements (ordered pairs).

4. Relations

Definition: A relation R from a set A to a set B is any subset of the Cartesian product A × B.
R ⊆ A × B

If (a, b) ∈ R, we say that 'a' is related to 'b', sometimes written as aRb.

Domain: The set of all first elements (a's) in the ordered pairs of R.
Codomain: The entire set B.
Range: The set of all second elements (b's) in the ordered pairs of R. The range is always a subset of the codomain.

Example:
Let A = {1, 2, 3} and B = {2, 4, 6}
A × B = {(1,2), (1,4), (1,6), (2,2), (2,4), (2,6), (3,2), (3,4), (3,6)}
Let's define a relation R as "is half of": R = {(a, b) | b = 2a}
R = {(1, 2), (2, 4), (3, 6)}

Domain(R) = {1, 2, 3}
Codomain(R) = {2, 4, 6}
Range(R) = {2, 4, 6}

5. Functions and their properties

Definition: A function (or mapping) f from a set A to a set B (written f: A → B) is a special type of relation in which every element in the domain (A) is paired with exactly one element in the codomain (B).

We write y = f(x), where x is the independent variable (from domain A) and y is the dependent variable (from codomain B).

Function vs. Relation:

All functions are relations, but not all relations are functions.
Vertical Line Test: If you can draw a vertical line on a graph that crosses the curve more than once, it is not a function.

Properties of Functions (Types):

Injective (One-to-One):
A function is injective if different inputs (x-values) always produce different outputs (y-values).
Formally: If f(x₁) = f(x₂), then x₁ = x₂.
Example: f(x) = 2x + 1 is injective.
Non-Example: f(x) = x² is not injective (since f(2) = 4 and f(-2) = 4).
Surjective (Onto):
A function is surjective if every element in the codomain (B) is the output for at least one input from the domain (A).
Formally: The Range is equal to the Codomain.
Example: f(x) = 2x + 1 (from R to R) is surjective, as it can produce any real number.
Bijective (One-to-One and Onto):
A function is bijective if it is both injective and surjective. Bijective functions have an inverse function.

Economic Functions:

Demand Function: Q_d = f(P). Quantity demanded is a function of price.
Cost Function: TC = f(Q). Total cost is a function of quantity produced.
Utility Function: U = f(X, Y). Utility is a function of the quantity of two goods consumed.

6. Basic logarithmic

Definition: The logarithm is the inverse operation to exponentiation.
The statement y = log_b(x) (read: "log base b of x") is equivalent to the statement b^y = x.

b is the base (must be b > 0 and b ≠ 1).
x is the argument (must be x > 0).

Types of Logarithms:

Common Logarithm: Base 10. Written as log(x).
Natural Logarithm: Base e (Euler's number ≈ 2.718). Written as ln(x). This is the most common log in economics.

Key Properties of Logarithms (Rules):

Product Rule: ln(a * b) = ln(a) + ln(b)
Quotient Rule: ln(a / b) = ln(a) - ln(b)
Power Rule: ln(aⁿ) = n * ln(a)
Base Rules: ln(e) = 1 and ln(1) = 0

Economic Application: Log-Linear Models

Logarithms are used to measure percentage changes and elasticity. If we have a demand function ln(Q) = a - b * ln(P), the coefficient 'b' is directly the price elasticity of demand.

7. Limit of a function

Intuitive Definition: The limit of a function f(x) as x approaches a value 'c' is the value that f(x) (or y) gets closer and closer to.
We write this as: lim_x→c f(x) = L

The limit is about the approaching value, not what happens at the value. The function doesn't even need to be defined at x = c.

One-Sided Limits:

Left-Hand Limit (LHL): The value f(x) approaches as x approaches 'c' from the left (from numbers smaller than c).
Written: lim_x→c⁻ f(x)
Right-Hand Limit (RHL): The value f(x) approaches as x approaches 'c' from the right (from numbers larger than c).
Written: lim_x→c⁺ f(x)

Existence of a Limit:

A limit L exists at x = c if and only if the left-hand limit equals the right-hand limit.
lim_x→c⁻ f(x) = lim_x→c⁺ f(x) = L
If LHL ≠ RHL, the limit does not exist (DNE) at x = c.

Properties of Limits:

If lim_x→c f(x) = L and lim_x→c g(x) = M:

Sum/Difference: lim [f(x) ± g(x)] = L ± M
Product: lim [f(x) * g(x)] = L * M
Quotient:lim [f(x) / g(x)] = L / M (provided M ≠ 0)

8. Continuity

A function is continuous if its graph is a single unbroken curve. You can draw it without lifting your pen from the paper.

Formal Definition: A function f(x) is continuous at a point x = c if it meets all three of the following conditions:

f(c) is defined. (The point exists)

lim_x→c f(x) exists. (The LHL = RHL)

lim_x→c f(x) = f(c). (The limit value is the same as the point's value)

Types of Discontinuity:

Removable Discontinuity (Hole): Condition 1 or 3 fails. A "hole" in the graph that could be "plugged."
Jump Discontinuity: Condition 2 fails (LHL ≠ RHL). The graph "jumps" from one level to another. (e.g., a "step" function for taxi fares).
Infinite Discontinuity (Asymptote): The limit approaches +∞ or -∞. (e.g., f(x) = 1/x at x = 0).