Mathematical Induction and Logarithmic Functions

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Questions and Answers

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.

False (B)

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).

<p>ln</p>
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Match the following functions with their representations:

<p>Exponential function = exp(ax) Natural logarithm function = ln(x) Logarithmic function for base b = logb(x) Euler's number = e</p>
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What is the principle of mathematical induction primarily used for?

<p>To establish the truth of several related propositions (B)</p>
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The induction base in mathematical induction is the last proposition in the series.

<p>False (B)</p>
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What is the statement of the triangle inequality?

<p>|x − z| ≤ |x − y| + |y − z|</p>
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The first proposition in mathematical induction is known as the ______.

<p>induction base</p>
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Match the following terms related to mathematical induction with their definitions:

<p>Induction Base = The first proposition that is proven Induction Step = Demonstrates that if one proposition holds, the next one does too Inductive Argument = The reasoning used in mathematical induction Infinite Series = A collection of propositions that depend on a parameter n</p>
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Which of the following options best describes the outcome of proving both the induction base and induction step?

<p>All related propositions are proven (C)</p>
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For any real numbers x, y, z, the expression |x − z| ≤ |x − y| + |y − z| is known as the triangle equality.

<p>False (B)</p>
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What is the central idea behind mathematical induction?

<p>To prove propositions for all natural numbers by validating the base case and the inductive step.</p>
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What does the notation ‘∃x ∈ A : P’ signify?

<p>There exists at least one element x in A such that P holds. (D)</p>
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The statement ‘∀x ∈ A, P’ means that there exists at least one element x in A such that property P is true.

<p>False (B)</p>
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If A = {2, 4, 6, 8, 10, 12} and property P is x ≥ 10, write the logical statement representing this condition.

<p>∃x ∈ A : x ≥ 10</p>
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The notation ‘∀x ∈ A, P’ asserts that for all elements x of the set A, the property P is _____ (true or false).

<p>true</p>
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Match the mathematical terms with their definitions:

<p>∃ = There exists at least one element satisfying a property ∀ = For all elements of a set, a property is true ¬ = Negation of a statement ∈ = Element belongs to a set</p>
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Which of the following statements is the negation of ‘∃x ∈ A : P’?

<p>¬∃x ∈ A : P (A)</p>
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The property ‘x ≥ 0’ holds true for all elements in the set A = {2, 4, 6, 8, 10, 12}.

<p>True (A)</p>
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Using the set B = {students enrolled in ECN115}, write a logical statement that reflects the attendance of students.

<p>∃x ∈ B : R</p>
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What is the value of $5!$?

<p>120 (A)</p>
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The binomial coefficient $C(n, j)$ is defined for all integers j and n.

<p>False (B)</p>
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What is the binomial formula for $(1 + x)^n$?

<p>(1 + x)^n = ∑_{j=0}^{n} C(n, j) x^j</p>
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The notation $k!$ denotes the ______ of the number k.

<p>factorial</p>
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Match the following terms with their definitions:

<p>Binomial coefficient = A specific type of coefficient defined for combinations Factorial = The product of an integer and all integers below it Function = A relation from a set of inputs to a set of possible outputs Domain = The set of possible inputs for a function</p>
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Which of the following statements is true regarding functions?

<p>Each element in the domain must be associated with a unique element in the range. (A)</p>
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The factorial of 0 is equal to 1.

<p>True (A)</p>
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Explain what a function is in your own words.

<p>A function is a mathematical relationship that assigns exactly one output for each input from a defined set.</p>
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Which of the following statements is true for the set A = {2, 4, 6, 8, 10, 12}?

<p>Not all x in A are greater than or equal to 10. (D)</p>
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The statement ∀x ∈ A, x ≥ 20 is true.

<p>False (B)</p>
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What is the equivalent statement of 'There exists a non-negative real number'?

<p>Not all real numbers are negative.</p>
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The absolute value of a negative number x is given by |x| = _____.

<p>-x</p>
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Match the following sets with their properties regarding rail usage:

<p>Set A: {bicycle, car, boat} = No vehicle uses rails Set B: {bus, tram, underground} = Some vehicles use rails Set C: {tram, overground, train} = All vehicles use rails</p>
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Which of the following statements is equivalent to '∀x ∈ C, x uses rails'?

<p>There is no vehicle in C that does not use rails. (C)</p>
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The statement '∃x ∈ A : not P' is equivalent to 'not ∀x ∈ A, P'.

<p>True (A)</p>
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What does the Triangle Inequality state about distances?

<p>The distance from x to z should not be longer than going from x to y and then y to z.</p>
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What is the first step in the process of mathematical induction?

<p>Establishing the induction base (D)</p>
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In mathematical induction, the induction hypothesis states that proposition Pn is true.

<p>True (A)</p>
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What does the induction step in mathematical induction demonstrate?

<p>It demonstrates that if Pn is true, then Pn+1 is also true.</p>
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The proposition that states the sum of the first n natural numbers is equal to _____ is a part of the mathematics of induction.

<p>n(n + 1)/2</p>
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Match the following components to their definitions in mathematical induction:

<p>Induction Base = The first proposition that needs to be proven Induction Step = Demonstrating Pn implies Pn+1 Induction Hypothesis = Assuming Pn is true Proposition Pn = The general statement for n natural numbers</p>
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Which of the following statements is true about mathematical induction?

<p>It uses the concept of an inductive step and base. (B)</p>
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In mathematical induction, proving all propositions means that only P1 needs to be verified.

<p>False (B)</p>
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What does the notation ∑ from j=1 to n represent in the context of induction?

<p>It represents the sum of the first n natural numbers.</p>
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Flashcards

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?

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?

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?

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?

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?

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?

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?

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

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

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)

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))

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)

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 (∀)

A statement that holds true for every element in a set.

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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|)

The distance of a number from zero, regardless of its sign. It's always non-negative.

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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

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

A formula that expands the power of a binomial (a + b) raised to a non-negative integer.

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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

Functions widely used in mathematics, featuring operations like addition, subtraction, multiplication, division, exponentiation, and roots.

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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

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

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)

The set of all positive integers: 1, 2, 3, 4...

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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

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)

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

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

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

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!)

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)

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)

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)

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)

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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Study Notes

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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