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Questions and Answers
What is the value of Euler’s number, commonly denoted as e?
What is the value of Euler’s number, commonly denoted as e?
- 1.61803
- 2.71828 (correct)
- 3.14159
- 0.57721
The function logb(x) is defined for all real numbers x.
The function logb(x) is defined for all real numbers x.
False (B)
What is the relationship expressed by the logarithmic function logb(x)?
What is the relationship expressed by the logarithmic function logb(x)?
b^y = x
When b equals Euler's number, the logarithmic function is expressed as _____(x).
When b equals Euler's number, the logarithmic function is expressed as _____(x).
Match the following functions with their representations:
Match the following functions with their representations:
What is the principle of mathematical induction primarily used for?
What is the principle of mathematical induction primarily used for?
The induction base in mathematical induction is the last proposition in the series.
The induction base in mathematical induction is the last proposition in the series.
What is the statement of the triangle inequality?
What is the statement of the triangle inequality?
The first proposition in mathematical induction is known as the ______.
The first proposition in mathematical induction is known as the ______.
Match the following terms related to mathematical induction with their definitions:
Match the following terms related to mathematical induction with their definitions:
Which of the following options best describes the outcome of proving both the induction base and induction step?
Which of the following options best describes the outcome of proving both the induction base and induction step?
For any real numbers x, y, z, the expression |x − z| ≤ |x − y| + |y − z| is known as the triangle equality.
For any real numbers x, y, z, the expression |x − z| ≤ |x − y| + |y − z| is known as the triangle equality.
What is the central idea behind mathematical induction?
What is the central idea behind mathematical induction?
What does the notation ‘∃x ∈ A : P’ signify?
What does the notation ‘∃x ∈ A : P’ signify?
The statement ‘∀x ∈ A, P’ means that there exists at least one element x in A such that property P is true.
The statement ‘∀x ∈ A, P’ means that there exists at least one element x in A such that property P is true.
If A = {2, 4, 6, 8, 10, 12} and property P is x ≥ 10, write the logical statement representing this condition.
If A = {2, 4, 6, 8, 10, 12} and property P is x ≥ 10, write the logical statement representing this condition.
The notation ‘∀x ∈ A, P’ asserts that for all elements x of the set A, the property P is _____ (true or false).
The notation ‘∀x ∈ A, P’ asserts that for all elements x of the set A, the property P is _____ (true or false).
Match the mathematical terms with their definitions:
Match the mathematical terms with their definitions:
Which of the following statements is the negation of ‘∃x ∈ A : P’?
Which of the following statements is the negation of ‘∃x ∈ A : P’?
The property ‘x ≥ 0’ holds true for all elements in the set A = {2, 4, 6, 8, 10, 12}.
The property ‘x ≥ 0’ holds true for all elements in the set A = {2, 4, 6, 8, 10, 12}.
Using the set B = {students enrolled in ECN115}, write a logical statement that reflects the attendance of students.
Using the set B = {students enrolled in ECN115}, write a logical statement that reflects the attendance of students.
What is the value of $5!$?
What is the value of $5!$?
The binomial coefficient $C(n, j)$ is defined for all integers j and n.
The binomial coefficient $C(n, j)$ is defined for all integers j and n.
What is the binomial formula for $(1 + x)^n$?
What is the binomial formula for $(1 + x)^n$?
The notation $k!$ denotes the ______ of the number k.
The notation $k!$ denotes the ______ of the number k.
Match the following terms with their definitions:
Match the following terms with their definitions:
Which of the following statements is true regarding functions?
Which of the following statements is true regarding functions?
The factorial of 0 is equal to 1.
The factorial of 0 is equal to 1.
Explain what a function is in your own words.
Explain what a function is in your own words.
Which of the following statements is true for the set A = {2, 4, 6, 8, 10, 12}?
Which of the following statements is true for the set A = {2, 4, 6, 8, 10, 12}?
The statement ∀x ∈ A, x ≥ 20 is true.
The statement ∀x ∈ A, x ≥ 20 is true.
What is the equivalent statement of 'There exists a non-negative real number'?
What is the equivalent statement of 'There exists a non-negative real number'?
The absolute value of a negative number x is given by |x| = _____.
The absolute value of a negative number x is given by |x| = _____.
Match the following sets with their properties regarding rail usage:
Match the following sets with their properties regarding rail usage:
Which of the following statements is equivalent to '∀x ∈ C, x uses rails'?
Which of the following statements is equivalent to '∀x ∈ C, x uses rails'?
The statement '∃x ∈ A : not P' is equivalent to 'not ∀x ∈ A, P'.
The statement '∃x ∈ A : not P' is equivalent to 'not ∀x ∈ A, P'.
What does the Triangle Inequality state about distances?
What does the Triangle Inequality state about distances?
What is the first step in the process of mathematical induction?
What is the first step in the process of mathematical induction?
In mathematical induction, the induction hypothesis states that proposition Pn is true.
In mathematical induction, the induction hypothesis states that proposition Pn is true.
What does the induction step in mathematical induction demonstrate?
What does the induction step in mathematical induction demonstrate?
The proposition that states the sum of the first n natural numbers is equal to _____ is a part of the mathematics of induction.
The proposition that states the sum of the first n natural numbers is equal to _____ is a part of the mathematics of induction.
Match the following components to their definitions in mathematical induction:
Match the following components to their definitions in mathematical induction:
Which of the following statements is true about mathematical induction?
Which of the following statements is true about mathematical induction?
In mathematical induction, proving all propositions means that only P1 needs to be verified.
In mathematical induction, proving all propositions means that only P1 needs to be verified.
What does the notation ∑ from j=1 to n represent in the context of induction?
What does the notation ∑ from j=1 to n represent in the context of induction?
Flashcards
What does the notation "∃x ∈ A : P" represent?
What does the notation "∃x ∈ A : P" represent?
The notation "∃x ∈ A : P" means that there exists at least one element 'x' in the set 'A', such that property 'P' holds for 'x'.
How is "̸ ∃x ∈ A : P" interpreted?
How is "̸ ∃x ∈ A : P" interpreted?
The statement "̸ ∃x ∈ A : P" means there does not exist any element 'x' in the set 'A' for which property 'P' is true.
What does the notation "∀x ∈ A, P" represent?
What does the notation "∀x ∈ A, P" represent?
The notation "∀x ∈ A, P" means that for every element 'x' in the set 'A', property 'P' is true.
How to prove a statement involving "For all" false?
How to prove a statement involving "For all" false?
A statement involving "For all" is false if there exists at least one element of the set that does not satisfy the property.
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What is the triangle inequality?
What is the triangle inequality?
It states that the distance between two points is less than or equal to the sum of the distances from each point to a third point along the same path.
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What is mathematical induction?
What is mathematical induction?
A method that uses a series of steps to prove a statement is true for all natural numbers.
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What is the binomial formula?
What is the binomial formula?
A formula that expands the power of a binomial (a + b)^n into a sum of terms, where n is a positive integer.
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What is a function?
What is a function?
A function is a rule that assigns to each input in its domain exactly one output in its co-domain.
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Exponential Function
Exponential Function
A mathematical function where the input is an exponent and the output is the result of raising a base number to that exponent.
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Logarithmic Function
Logarithmic Function
A mathematical function that undoes an exponential function. It finds the exponent needed to raise a base to a given value.
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Euler's Number (e)
Euler's Number (e)
The base of the natural logarithm, approximately equal to 2.71828. It is a fundamental constant in mathematics.
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Natural Logarithm (ln(x))
Natural Logarithm (ln(x))
A special case of the logarithmic function where the base is Euler's number (e). It is denoted as ln(x).
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logb(x)
logb(x)
Used to denote the logarithmic function to any base. For example, log2(x) represents the logarithm of x base 2.
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Universal Quantifier (∀)
Universal Quantifier (∀)
A statement that holds true for every element in a set.
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Existential Quantifier (∃)
Existential Quantifier (∃)
A statement that confirms the existence of at least one element in a set that satisfies a specific property.
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Absolute Value (|x|)
Absolute Value (|x|)
The distance of a number from zero, regardless of its sign. It's always non-negative.
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Triangle Inequality
Triangle Inequality
The sum of the absolute values of two sides of a triangle is always greater than or equal to the absolute value of the third side.
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Mathematical Induction
Mathematical Induction
A formal proof technique where you show that a statement is true for a base case and then prove that if it's true for one case, it's also true for the next.
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Binomial Formula
Binomial Formula
A formula that expands the power of a binomial (a + b) raised to a non-negative integer.
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Function
Function
A rule that assigns each element in a set (domain) to a unique element in another set (codomain).
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Common Numerical Functions
Common Numerical Functions
Functions widely used in mathematics, featuring operations like addition, subtraction, multiplication, division, exponentiation, and roots.
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Induction Base
Induction Base
The first proposition (P1) in a mathematical induction proof. It establishes the truth of the statement for the smallest value of 'n'.
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Induction Step
Induction Step
The proposition that proves if a statement holds true for a specific value of 'n', it also holds true for the next consecutive value (n+1). This establishes the pattern for the entire chain.
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Induction Hypothesis
Induction Hypothesis
A statement assumed true in the induction step. It implies that the formula holds for the current value of 'n'.
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Natural Numbers (N)
Natural Numbers (N)
The set of all positive integers: 1, 2, 3, 4...
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Sum of First 'n' Natural Numbers
Sum of First 'n' Natural Numbers
The sum of the first 'n' natural numbers can be calculated using the formula n(n+1)/2.
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Proposition Pn
Proposition Pn
The proposition that states that the sum of the first 'n' natural numbers is equal to n(n+1)/2.
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P1 (Induction Base Example)
P1 (Induction Base Example)
The specific instance of Proposition Pn when n=1. It states that the sum of the first natural number (1) is equal to 1(1+1)/2, which is 1.
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Mathematical Induction Proof
Mathematical Induction Proof
The proof technique that consists of demonstrating both the induction base and the induction step to guarantee the truth of all propositions Pn for all values of n.
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Establishing the Base Case
Establishing the Base Case
The process of establishing that a claim is true for the first case, often n=1, in Mathematical Induction.
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Proving the Induction Step
Proving the Induction Step
The process of proving that the claim holds for any subsequent case given that it holds for the previous case (n=k implies n=k+1).
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Factorial (k!)
Factorial (k!)
A number, often denoted as k!, which is the product of all positive integers less than or equal to k. It is defined recursively: 0! = 1, k! = (k-1)! * k.
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Binomial Coefficient (nCj)
Binomial Coefficient (nCj)
A coefficient used in the binomial theorem, calculated as n! / (j! * (n-j)!). It represents the number of ways to choose j items from a set of n items.
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Function (f: D -> R)
Function (f: D -> R)
A formal statement that associates each element of a domain (input set) with a unique element of a range (output set). It can be represented as a set of ordered pairs, where each input maps to a single, specific output.
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Domain (D)
Domain (D)
The set of all acceptable input values for a function. It is the starting point for the function's rule.
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Range (R)
Range (R)
The set of all possible output values that a function can produce. It's the result of applying the function's rule to the domain.
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Lecture 2: Mathematical Methods in Economics and Finance
- Quiz I: Scheduled for Monday, October 14th.
- Quiz Scope: Covers material from the first two lectures and homeworks (up to the end of the third week).
- Quiz Weighting: 7.5% of the total grade.
- Quiz Duration: 1 hour, can be started at any time on the due date (October 14th).
- Instructions: Students must work alone. Each student receives a unique version of the quiz questions.
- Question Types: Students will answer questions numerically, select single-choice options, or choose all correct answers (with potential for one or more correct answers).
- Question Count: Between 5 and 10 questions.
Outline
- Quantifiers: "There exists" and "For all"
- Absolute Value and Triangle Inequality: Defined and illustrated.
- Mathematical Induction: A method for proving multiple similar propositions based on a parameter (n∈N).
- Induction Base: Establishing the first proposition (P1)
- Induction Step: Proving the relation between propositions at any index (Pn) and the next index (Pn+1).
- All propositions can be proven true once base and step are proven.
- Binomial Formula: Expansion of (x + y)n .
- Functions: Definitions: A function takes each element of a domain (D) set to an element of a range (R) set.
- Common Numerical Functions: Includes integer powers, inverse natural powers, rational powers, polynomials.
"There Exists" and "For All"
- ∃x ∈ A : P: There exists an element x in set A such that property P holds.
- ∀x ∈ A : P: For all elements x in set A, property P holds.
Absolute Value and Triangle Inequality6
- The absolute value of a real number x, |x|, is a distance on a number line.
- Triangle Inequality: Given three points x, y and z on a number line: |x - z| ≤ |x - y| + |y - z|.
Mathematical Induction
- A proof technique used for multiple, similar statements or propositions that depend on a parameter n.
- The proof involves two steps: a base case and a recursive step.
Binomial Formula
- The Binomial Theorem states that (x + y)<sup>n</sup> = Σ (n choose k) x<sup>n-k</sup> y<sup>k</sup>, with k from 0 to n, illustrating the coefficients as binomial coefficients representing selection ways.
Functions: Definitions
- Definition of a function: a mapping between two sets in which each element in the first set is associated with a unique element in the second set.
Common Numerical Functions (Example)
- Integer Powers: f(x)=xk where k ∈ N.
- Inverse Natural Powers: f(x) = x1/n
- Rational Powers: f(x) = xp/q
- Polynomials: f(x)= Σakxk
- Exponential Functions: f(x)=bax with b > 0 and a,b ∈ R.
- Logarithmic Functions: f(x)= logbx with b > 0 and b ≠ 1
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