Set Theory and Rational Numbers Quiz
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Questions and Answers

What property must a nonempty subset I of N satisfy to be considered an initial segment?

  • If n ∈ I and m ≤ n, then m ∈ I. (correct)
  • If n ∈ I, then all integers greater than n are also in I.
  • If n ∈ I and m = n, then m ∈ I.
  • If n ∈ I and m ≥ n, then m ∈ I.

    • Which of the following describes a bijective mapping?

  • A mapping that is one to one but not onto.
  • A mapping that exists only for finite sets.
  • A mapping that is onto but not one to one.
  • A mapping that is both one to one and onto. (correct)

    • In a finite set A, which statement is true?

  • A is either empty or has a finite number of elements. (correct)
  • There exists an infinite number of elements without repetition.
  • A can have a repeating sequence of elements.
  • There exists a mapping c: In → A that is not bijective.

    • What happens if m.n = 0 in multiplication?

    <p>At least one of m or n must be zero.</p> Signup and view all the answers

    Which of the following describes the operation defined on the set of rational numbers Q?

    <p>The operation is addition as described by p r ps + qr.</p> Signup and view all the answers

    What can be concluded if n is not an element of an initial segment I and n is greater than some element m in I?

    <p>I does not cover all integers greater than or equal to m.</p> Signup and view all the answers

    What is represented by the set Q∗?

    <p>The set of rational numbers excluding both 0 and 1.</p> Signup and view all the answers

    Which statement about the cancellation property in multiplication is correct?

    <p>If m.n = p.n, then m and p must be equal only if n ≠ 0.</p> Signup and view all the answers

    What distinguishes finite sets from infinite sets?

    <p>Finite sets can be counted and listed without repetition.</p> Signup and view all the answers

    For every p in the set A = {p ∈ Q : p^2 < 2}, what can be concluded regarding the rational number q derived from p?

    <p>q will be greater than p.</p> Signup and view all the answers

    Which of the following showcases the commutativity property in multiplication?

    <p>n.m = m.n</p> Signup and view all the answers

    What operation does the proposition suggest is defined in (Q∗ , .)?

    <p>It defines multiplication on Q.</p> Signup and view all the answers

    Which statement is true about the lower bound a for the set B = {p ∈ Q : p^2 > 2}?

    <p>a is a lower bound but not the infimum if β &gt; a.</p> Signup and view all the answers

    In the context of rational numbers, what does the equation q = p - (p^2 - 2)/(p + 2) signify?

    <p>It defines a method to find a lower bound.</p> Signup and view all the answers

    When it is stated that for p in A, q^2 < 2, what does this imply about q?

    <p>q is a rational number less than 2.</p> Signup and view all the answers

    How is the multiplication of two elements in the set of rational numbers defined?

    <p>As pr/qs where p/q and r/s are rational numbers.</p> Signup and view all the answers

    What does it mean for a set D to be dense in the set of real numbers?

    <p>Every open interval contains a member of D.</p> Signup and view all the answers

    Which theorem states that there is a rational number p/q such that a < p/q < b for any a < b?

    <p>The Density Theorem.</p> Signup and view all the answers

    Which of the following statements is true regarding the set of rational numbers?

    <p>It may be bounded above but lacks a rational supremum.</p> Signup and view all the answers

    What is implied if a nonempty set S of real numbers is bounded above?

    <p>There exists a unique real number β that is the supremum of S.</p> Signup and view all the answers

    If p is the smallest integer such that p > qa, what is a key relationship derived from this?

    <p>qa &lt; p &lt; qa + q(b − a).</p> Signup and view all the answers

    What conclusion can be drawn from the Archimedean property when comparing ρ and ε?

    <p>Every positive integer n can be found such that nε &gt; ρ.</p> Signup and view all the answers

    Which expression can be derived for the set A defined as {p ∈ Q : p^2 < 2}?

    <p>p &lt; √2.</p> Signup and view all the answers

    When stating that rational numbers are not complete, this implies which characteristic?

    <p>There exists a bound but no least upper bound that is rational.</p> Signup and view all the answers

    What condition must be satisfied for a sequence {sn} to be considered a Cauchy sequence?

    <p>For every $ orall eta &gt; 0$, there exists an integer K such that $|s_n - s_m| &lt; eta$ if $m, n oldsymbol{≥} K$.</p> Signup and view all the answers

    What does the theorem regarding the limits of a sequence say about lim sup and lim inf?

    <p>Lim sup and lim inf must be equal for the limit to exist at a finite value.</p> Signup and view all the answers

    What happens when {sn} is a Cauchy sequence of real numbers?

    <p>The sequence is bounded.</p> Signup and view all the answers

    Under what condition does a sequence {sn} converge according to Cauchy's convergence criterion?

    <p>If there exists an integer N such that $|s_n - s_m| &lt; ε$ for m, n ≥ N.</p> Signup and view all the answers

    What is the significance of the values of s and s when a sequence converges?

    <p>s and s must be finite and equal for the limit to exist.</p> Signup and view all the answers

    What is NOT a requirement for a sequence to converge based on the definitions given?

    <p>The sequence must be strictly increasing.</p> Signup and view all the answers

    Which statement accurately reflects the relation between limit superior, limit inferior, and the convergence of a sequence?

    <p>If both limits exist and are equal, the sequence converges.</p> Signup and view all the answers

    What does the term 'bounded sequence' imply in the context of a Cauchy sequence?

    <p>The terms of the sequence are limited to a certain range.</p> Signup and view all the answers

    What is the purpose of proving that $y_n = x$?

    <p>To illustrate that both inequalities $y_n &lt; x$ and $y_n &gt; x$ lead to a contradiction</p> Signup and view all the answers

    What does the inequality $|a + b| ≤ |a| + |b|$ indicate about real numbers?

    <p>It applies universally regardless of the signs of $a$ and $b$.</p> Signup and view all the answers

    In case (c) of the proof of $|a + b| ≤ |a| + |b|$, what condition is satisfied?

    <p>$a ≥ 0$ and $b ≤ 0$</p> Signup and view all the answers

    What does the corollary $|a - b| ≥ |a| - |b|$ signify?

    <p>It establishes a lower bound for the absolute difference of two numbers.</p> Signup and view all the answers

    What conclusion can be drawn from the assumption that $y < x$?

    <p>It creates an upper bound that contradicts $y$ being the least upper bound.</p> Signup and view all the answers

    Which of the following conditions applies when proving $|a + b|$ for cases (a) and (b)?

    <p>Both $a$ and $b$ are positive.</p> Signup and view all the answers

    What does the identity $b^n - a^n = (b - a)nb^{n-1}$ imply when $0 < a < b$?

    <p>It establishes a relationship between power functions of different bases.</p> Signup and view all the answers

    What contradiction arises from the statement $y - k < y$ in the context of upper bounds?

    <p>It shows that $y - k$ cannot exceed $y$, contradicting the definition of upper bounds.</p> Signup and view all the answers

    What property is used to define the supremum of a nonempty set of bounded real numbers?

    <p>It is the least upper bound that satisfies certain conditions.</p> Signup and view all the answers

    According to the completeness axiom, what can be said about a nonempty set of real numbers that is bounded above?

    <p>It must have a supremum that is a real number.</p> Signup and view all the answers

    What defines the first property of the supremum?

    <p>It is less than or equal to every element in the set.</p> Signup and view all the answers

    What does the second property of the supremum state regarding any number less than beta?

    <p>There exists an element in the set that exceeds it.</p> Signup and view all the answers

    Which of the following statements is true about the uniqueness of the supremum?

    <p>If one supremum exists, no other can satisfy both properties.</p> Signup and view all the answers

    What does the Archimedean property imply about two positive numbers, ρ and ε?

    <p>There exists an integer n such that nε &gt; ρ.</p> Signup and view all the answers

    In proving the Archimedean property, what assumption is made if the statement is assumed false?

    <p>ρ is an upper bound of the set S where S = {x = nε, n ∈ Z}.</p> Signup and view all the answers

    What can be concluded about a real number that is both an upper bound and satisfies the second property for a given set S?

    <p>It is the supremum of S.</p> Signup and view all the answers

    Study Notes

    Course Information

    • Course title: Real Analysis I (MTH621)
    • Instructor: Salman Amin Malik
    • Number of credit hours: 3
    • Textbooks:
      • W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, 1976. ISBN: 9780070542358.
      • W. F. Trench, Introduction to Real Analysis, Pearson Education, 2013.

    Course Objectives

    • Provide a rigorous treatment of fundamental mathematical analysis concepts.
    • Builds on calculus knowledge (single and multivariable) and differential equations.

    Learning Outcomes

    • Understand set theoretic statements, real and complex number systems.
    • Apply mathematical induction, discuss ordered sets of sequences and series.
    • Define the limit of a function and prove theorems about limits and functions.
    • Evaluate continuity of real-valued functions and relevant theorems.
    • Understand derivatives of functions and relevant theorems about differentiability.
    • Prove and apply Bolzano-Weierstrass and Mean value theorems.
    • Define the Riemann integral and prove related results.

    Prerequisites

    • Calculus with Analytical Geometry

    Topics Covered

    • The Real Number System
      • Basic set theory
      • Number theory
      • Principle of Mathematical Induction
      • Finite and Infinite Sets
      • The Set of Rational Numbers
      • Ordered Sets
      • Least Upper Bound Property / Completeness Axiom
      • The Archimedean Property
      • Dense Sets in R
    • Sequences and Series
      • Sequences
      • Series
      • Convergence
    • Continuity
      • Limits
      • One Sided Limits
      • Continuity
      • Uniform Continuity
    • Differentiability
      • Derivative
      • One-sided derivatives
      • Differentiability implies Continuity
      • The mean value theorem
    • Riemann Integration
      • Riemann Sums
      • Upper and Lower Integrals
      • Fundamental Theorem of Calculus
      • Integration by Parts
      • Integration by Substitution

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    Description

    Test your knowledge on initial segments, bijective mappings, and properties of rational numbers. This quiz covers essential concepts in set theory, multiplication, and the characteristics that differentiate finite and infinite sets. Perfect for students looking to strengthen their understanding of these mathematical principles.

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