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

    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> Signup and view all the answers

    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> Signup and view all the answers

    What is the principle of mathematical induction primarily used for?

    <p>To establish the truth of several related propositions</p> Signup and view all the answers

    The induction base in mathematical induction is the last proposition in the series.

    <p>False</p> Signup and view all the answers

    What is the statement of the triangle inequality?

    <p>|x − z| ≤ |x − y| + |y − z|</p> Signup and view all the answers

    The first proposition in mathematical induction is known as the ______.

    <p>induction base</p> Signup and view all the answers

    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> Signup and view all the answers

    Which of the following options best describes the outcome of proving both the induction base and induction step?

    <p>All related propositions are proven</p> Signup and view all the answers

    For any real numbers x, y, z, the expression |x − z| ≤ |x − y| + |y − z| is known as the triangle equality.

    <p>False</p> Signup and view all the answers

    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> Signup and view all the answers

    What does the notation ‘∃x ∈ A : P’ signify?

    <p>There exists at least one element x in A such that P holds.</p> Signup and view all the answers

    The statement ‘∀x ∈ A, P’ means that there exists at least one element x in A such that property P is true.

    <p>False</p> Signup and view all the answers

    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> Signup and view all the answers

    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> Signup and view all the answers

    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> Signup and view all the answers

    Which of the following statements is the negation of ‘∃x ∈ A : P’?

    <p>¬∃x ∈ A : P</p> Signup and view all the answers

    The property ‘x ≥ 0’ holds true for all elements in the set A = {2, 4, 6, 8, 10, 12}.

    <p>True</p> Signup and view all the answers

    Using the set B = {students enrolled in ECN115}, write a logical statement that reflects the attendance of students.

    <p>∃x ∈ B : R</p> Signup and view all the answers

    What is the value of $5!$?

    <p>120</p> Signup and view all the answers

    The binomial coefficient $C(n, j)$ is defined for all integers j and n.

    <p>False</p> Signup and view all the answers

    What is the binomial formula for $(1 + x)^n$?

    <p>(1 + x)^n = ∑_{j=0}^{n} C(n, j) x^j</p> Signup and view all the answers

    The notation $k!$ denotes the ______ of the number k.

    <p>factorial</p> Signup and view all the answers

    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> Signup and view all the answers

    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.</p> Signup and view all the answers

    The factorial of 0 is equal to 1.

    <p>True</p> Signup and view all the answers

    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> Signup and view all the answers

    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.</p> Signup and view all the answers

    The statement ∀x ∈ A, x ≥ 20 is true.

    <p>False</p> Signup and view all the answers

    What is the equivalent statement of 'There exists a non-negative real number'?

    <p>Not all real numbers are negative.</p> Signup and view all the answers

    The absolute value of a negative number x is given by |x| = _____.

    <p>-x</p> Signup and view all the answers

    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> Signup and view all the answers

    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.</p> Signup and view all the answers

    The statement '∃x ∈ A : not P' is equivalent to 'not ∀x ∈ A, P'.

    <p>True</p> Signup and view all the answers

    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> Signup and view all the answers

    What is the first step in the process of mathematical induction?

    <p>Establishing the induction base</p> Signup and view all the answers

    In mathematical induction, the induction hypothesis states that proposition Pn is true.

    <p>True</p> Signup and view all the answers

    What does the induction step in mathematical induction demonstrate?

    <p>It demonstrates that if Pn is true, then Pn+1 is also true.</p> Signup and view all the answers

    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> Signup and view all the answers

    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> Signup and view all the answers

    Which of the following statements is true about mathematical induction?

    <p>It uses the concept of an inductive step and base.</p> Signup and view all the answers

    In mathematical induction, proving all propositions means that only P1 needs to be verified.

    <p>False</p> Signup and view all the answers

    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> Signup and view all the answers

    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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    Queen Mary University of London ECN115 Lecture 2 PDF

    Description

    Test your knowledge on mathematical induction and the properties of logarithmic functions. This quiz covers key concepts, definitions, and relationships, including Euler’s number and the triangle inequality. Challenge yourself to match terms and understand the principles behind these mathematical topics.

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