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Questions and Answers
What happens to the inequality if both sides are multiplied by a positive number?
What happens to the inequality if both sides are multiplied by a positive number?
If a > b and c < 0, then a · c > b · c.
If a > b and c < 0, then a · c > b · c.
False
What is the Least Upper Bound property of real numbers?
What is the Least Upper Bound property of real numbers?
If a non-empty subset of real numbers has an upper bound, then it has a least upper bound.
The closed interval ______ represents the set {x ∈ R | ___ ≤ x ≤ ___}.
The closed interval ______ represents the set {x ∈ R | ___ ≤ x ≤ ___}.
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Which of the following represents an open interval?
Which of the following represents an open interval?
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Match the interval notation with its description:
Match the interval notation with its description:
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An infinite interval can include negative infinity.
An infinite interval can include negative infinity.
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What does the notation (−∞, b] represent?
What does the notation (−∞, b] represent?
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What is the difference of the sets {apples, oranges, blueberries} and {bananas, oranges}?
What is the difference of the sets {apples, oranges, blueberries} and {bananas, oranges}?
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The set difference A∖B can contain elements from set B.
The set difference A∖B can contain elements from set B.
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What notation represents the set of natural numbers?
What notation represents the set of natural numbers?
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What percentage of the final assessment does the final exam constitute?
What percentage of the final assessment does the final exam constitute?
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Students can work in groups on the graded problem sets.
Students can work in groups on the graded problem sets.
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For any integer n, the property states that n + 1 is still in the set of natural numbers, which is known as the __________ property.
For any integer n, the property states that n + 1 is still in the set of natural numbers, which is known as the __________ property.
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Match the following number sets with their definitions:
Match the following number sets with their definitions:
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What mathematical concept involves finding maximum or minimum values given constraints?
What mathematical concept involves finding maximum or minimum values given constraints?
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The module includes themes such as sets, numbers, Mathematical Induction, and the ______ formula.
The module includes themes such as sets, numbers, Mathematical Induction, and the ______ formula.
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Which property states that a + b = b + a?
Which property states that a + b = b + a?
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Match the following types of assessment with their correct weightings:
Match the following types of assessment with their correct weightings:
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The distributive law states that a · (b + c) = a · b + a · c.
The distributive law states that a · (b + c) = a · b + a · c.
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When is Quiz I scheduled to take place?
When is Quiz I scheduled to take place?
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If A is a subset of B, what does A∖B equal?
If A is a subset of B, what does A∖B equal?
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The office hours for the lecturer are clearly specified.
The office hours for the lecturer are clearly specified.
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What is the lecturer's name for the ECN115 module?
What is the lecturer's name for the ECN115 module?
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What is the main purpose of the diagnostic quiz scheduled for September 27?
What is the main purpose of the diagnostic quiz scheduled for September 27?
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Weekly homeworks are graded and contribute to the final exam score.
Weekly homeworks are graded and contribute to the final exam score.
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Name one topic that will be covered next week.
Name one topic that will be covered next week.
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A __________ is a proposition that is accepted without proof.
A __________ is a proposition that is accepted without proof.
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Match the following terms with their correct definitions:
Match the following terms with their correct definitions:
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How many questions are in the diagnostic quiz and what is the duration?
How many questions are in the diagnostic quiz and what is the duration?
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Solutions to the homework problems will be explained during classes.
Solutions to the homework problems will be explained during classes.
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What is the final exam's percentage weight in the grading system?
What is the final exam's percentage weight in the grading system?
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What does property P1 of the summation operator state?
What does property P1 of the summation operator state?
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The property P2 of the summation operator confirms that the multiplication of a constant with a sum can be factored out.
The property P2 of the summation operator confirms that the multiplication of a constant with a sum can be factored out.
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What is the primary use of mathematical induction?
What is the primary use of mathematical induction?
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In the summation operator, $
∑_{j=k}^{n} (a_j + b_j)$ is equal to ______.
In the summation operator, $ ∑_{j=k}^{n} (a_j + b_j)$ is equal to ______.
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Match the properties of the summation operator with their descriptions:
Match the properties of the summation operator with their descriptions:
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Which of the following statements is true about property P3 of the summation operator?
Which of the following statements is true about property P3 of the summation operator?
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Mathematical induction can only be applied to even-numbered propositions.
Mathematical induction can only be applied to even-numbered propositions.
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What are the main components of the next week’s material outline?
What are the main components of the next week’s material outline?
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What is the first step in mathematical induction?
What is the first step in mathematical induction?
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In the induction step, it is sufficient to prove that Pn implies Pn+1 for only one value of n.
In the induction step, it is sufficient to prove that Pn implies Pn+1 for only one value of n.
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What is the term for the statement that proposition Pn is true when proving the induction step?
What is the term for the statement that proposition Pn is true when proving the induction step?
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The induction step states that if Pn is true, then __________ is also true.
The induction step states that if Pn is true, then __________ is also true.
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Match the following components of mathematical induction:
Match the following components of mathematical induction:
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What is the main purpose of mathematical induction?
What is the main purpose of mathematical induction?
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Mathematical induction can only be applied when the propositions depend on a single variable.
Mathematical induction can only be applied when the propositions depend on a single variable.
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How can you describe the relationship among the propositions P1, P2, P3, ..., Pn in the context of mathematical induction?
How can you describe the relationship among the propositions P1, P2, P3, ..., Pn in the context of mathematical induction?
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Study Notes
Module Information
- Module Code: ECN115
- Module Title: Mathematical Methods in Economics and Finance
- Lecturer: Evgenii Safonov
- Email: [email protected]
- Office: GC332
- Office Hours: TBA
- Lectures: Wednesdays 9:00-11:00 in GCG10 Peston LT, Thursdays 12:00-14:00 in GCG10 Peston LT
- Classes: See My Timetable on QMplus
Teaching Assistants
- Mahmoud Shahin: [email protected]
- Zhenghao Liu: [email protected]
- Tallon Trent Howie: [email protected]
Module Content
- Refreshes existing mathematical knowledge and introduces mathematical tools used in undergraduate economics and finance.
- Themes:
- Introduction: sets, numbers, mathematical induction, binomial formula
- Elements of one-variable mathematical analysis and calculus
- Constrained optimisation (one variable)
- Introduction to multivariate analysis and optimisation
- (If time permits) other topics including linear algebra
Assessment
- Three types:
- Online quizzes (15%): Individual work
- Graded problem sets (15%): Individual solutions, group discussions allowed
- Final exam (70%): Individual work, centrally administered
- Timing:
- Quiz 1 (7.5%): One hour, October 14th, choose time between 00:00 to 23:59
- Problem Set 1(7.5%): Available 1 week in advance or earlier, November 3rd
- Quiz 2 (7.5%): One hour, December 2nd; choose time between 00:00 to 23:59
- Problem Set 2 (7.5%): Available 1 week in advance or earlier, December 10th
- Final exam (70%): Two hours, January (paper and pen only)
Important Component
- Weekly homeworks (not graded): Essential for practice.
- A mix of basic and advanced problems.
- Solutions are posted online
Diagnostic Quiz
- Friday, September 27th
- Does not count towards the grade
- Objective: to assess background level in mathematics
- Format: 20 questions (mostly multiple choice, some requiring calculations)
- Duration: 80 minutes
Last Year's Materials
- Previous exam topics and exams from the past 2 years are available on QM+: ECN115 -> Key Module Information -> Syllabus
- Previous year's exam materials might be slightly different from this year's
- Solutions will be posted closer to the exams
Outline
- Elementary logic
- Sets
- Number systems and properties of real numbers
- Basic inequalities
- Summation operator
- Next week's material: Mathematical Induction
Sets
- A set is a collection of objects
- Elements are objects within a set denoted by ∈
- Two sets (A and B) are equal if each element in A is in B and each element in B is in A (A=B).
- An empty set is a set with no objects denoted by Ø or {}.
- A subset is a set where every element is also contained within another set, indicating a relationship of inclusion; if set A is a subset of set B, we denote this as A ⊆ B.
- Conversely, a superset is one that includes all elements of a particular subset; thus, if set B contains all elements of set A, we can say B ⊇ A.
- For example, if we have set A = {1, 2} and set B = {1, 2, 3, 4}, then A is a subset of B while B is a superset of A.
- Understanding these concepts is fundamental in set theory, as they help categorize and understand the relationships between different sets.
- Not all concepts can be defined as sets, as showcased by Russell's paradox. This highlights a critical flaw in naive set theory, revealing contradictions arising when a set includes itself.
- Set relations and operations are fundamental concepts in set theory. Subsets refer to a set where all its elements are contained within another set. The intersection (∩) of two sets includes only the elements common to both sets. The union (∪) combines all unique elements from both sets. The difference (∖) illustrates the elements that belong to one set but not another, providing insight into the relationships between sets.
- This notation denotes a set that comprises all elements, denoted by x, which are derived from set A and simultaneously satisfy a specific condition or property, referred to as property P, thereby creating a defined subset with distinct characteristics.
- Set subtraction( A∖B)
Number Systems
- Natural numbers (N): {1, 2, 3,...}
- Integers (Z): {... -3, -2, -1, 0, 1, 2, 3,...}
- Rational numbers (Q): All expressions p/q where p ∈ Z and q ∈ N (q≠0)
- Real numbers (R): The set of points on the number line
Properties of Real Numbers
- Order Relation: In mathematics, an order relation provides a way to compare elements within a set. This could include total orders, where every pair of elements can be compared, or partial orders, where some elements may not be directly comparable. Understanding these concepts is critical in various fields, including analysis and algebra, as they lay the groundwork for discussing concepts like convergence and limits.
- Addition and Multiplication operations: These fundamental operations in mathematics exhibit several key properties. The commutative property states that changing the order of the numbers involved does not change the sum or product (e.g., a + b = b + a and ab = ba). The associative property indicates that the grouping of numbers does not affect their sum or product (e.g., (a + b) + c = a + (b + c) and (ab)c = a(bc)). Furthermore, every number has an additive inverse (a number that, when added to the original, results in zero) and a multiplicative inverse (a number that, when multiplied by the original, results in one). Lastly, the identity elements of addition (0) and multiplication (1) are crucial, as they do not alter the original value of a number when used in their respective operations.
- Least Upper Bound property: Also known as the supremum property, this concept asserts that any non-empty set of real numbers that is bounded above has a least upper bound (supremum) in the real numbers. This property is essential in real analysis, ensuring that certain limits and bounds exist, thus providing a foundation for discussing continuity, compactness, and convergence in calculus and advanced mathematical theories.
Intervals of Real Numbers
- Closed interval [a, b] = {x ∈ R | a ≤ x ≤ b}
- Open interval (a, b) = {x ∈ R | a < x < b}
- Half-open/half-closed intervals ((a, b], [a, b))
Basic Inequalities
- The properties of real numbers, such as the distributive, associative, and commutative properties, play a crucial role in solving inequalities. Understanding how to manipulate these properties helps in rearranging and simplifying expressions. Additionally, techniques such as graphing, interval testing, and using the number line can effectively identify the solution sets for various types of inequalities.
- Basic techniques for solving inequalities often include isolating the variable, applying inverse operations, and carefully considering the impact of multiplying or dividing by negative numbers, which reverses the inequality sign. These foundational approaches are essential for ensuring accurate solutions in mathematical contexts involving inequalities.
- Example problems
Summation Operator
- Summation notation (Σ)
- Properties of summation:
- (P1)
- (P2)
- (P3)
- Examples: calculations with the summation operator
Mathematical Induction
- A technique for proving statements that depend on a natural number (n)
- Requires a proven induction base (first case ) and induction step
Additional Notes
- The lecture notes cover a range of mathematical topics relevant to economics and finance.
- The module emphasizes understanding the underlying principles behind calculations, rather than just mechanical application of formulas.
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Description
Test your understanding of the key mathematical concepts outlined in the ECN115 module. This quiz covers sets, calculus, constrained optimization, and introduces multivariate analysis. Perfect for students looking to refresh their mathematical knowledge essential for economics and finance.
