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Questions and Answers

What characterizes a sequence as strictly increasing?

  • an+1 > an for all n ∈ N (correct)
  • an+1 < an for all n ∈ N
  • an+1 = an for any n ∈ N
  • an is constant for all n ∈ N

    • Which of the following sequences is monotonic?

  • an = sin(n)
  • an = n + 2 (correct)
  • an = n^2 (correct)
  • an = (-1)^n

    • What is the definition of a sequence that is bounded above?

  • an approaches infinity as n approaches infinity
  • There exists M ∈ R such that an ≤ M for all n ∈ N (correct)
  • an < 0 for all n ∈ N
  • There exists M ∈ R such that an ≥ M for all n ∈ N

    • Which condition indicates that a sequence is not bounded below?

    <p>The sequence has terms that can go below any selected M</p> Signup and view all the answers

    What is an example of a strictly decreasing sequence?

    <p>an = -n</p> Signup and view all the answers

    In the context of the given definitions, which statement is true about the sequence an = -n?

    <p>All of the above</p> Signup and view all the answers

    What is meant by a monotonic sequence?

    <p>A sequence that is either strictly increasing or strictly decreasing</p> Signup and view all the answers

    Which of the following characteristics does NOT define a bounded sequence?

    <p>The sequence oscillates infinitely</p> Signup and view all the answers

    What is the condition for a sequence (an) to be classified as bounded?

    <p>There exists M ∈ R such that for all n ∈ N, |an| ≤ M.</p> Signup and view all the answers

    Which of the following sequences is an example of a bounded sequence?

    <p>an = 1/n</p> Signup and view all the answers

    What does it mean for a sequence (an) to be unbounded?

    <p>For every real number M, there exists an n such that |an| &gt; M.</p> Signup and view all the answers

    If a sequence is both bounded above and bounded below, what can we deduce about the sequence?

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

    In terms of logic, how can the statement 'a sequence (an) is unbounded' be formulated?

    <p>It is not true that (an) is bounded above and bounded below.</p> Signup and view all the answers

    Why can we not determine boundedness for a sequence of complex numbers using inequalities?

    <p>Complex numbers cannot be ordered in the same way as real numbers.</p> Signup and view all the answers

    Which of the following best describes the sequence an = -n?

    <p>It is unbounded.</p> Signup and view all the answers

    What is necessary for a sequence (an) of complex numbers to be considered bounded?

    <p>The absolute values |an| must all be less than some constant M.</p> Signup and view all the answers

    What must be true for a subset of complex numbers A to be considered bounded?

    <p>There exists an M ∈ R such that for all z ∈ A, |z| ≤ M.</p> Signup and view all the answers

    If a set A of complex numbers is bounded above, what can be inferred about its elements?

    <p>There exists a real number M1 such that all z ∈ A satisfy z ≤ M1.</p> Signup and view all the answers

    Which inequality correctly describes the boundedness of a set A that is both bounded above and below?

    <p>-M ≤ x ≤ M for all x ∈ A.</p> Signup and view all the answers

    What contradiction arises when proving that A = {z ∈ C | Re(z) ≤ 1} is unbounded?

    <p>We assumed there exists M such that |z| ≤ M but found a z where |z| &gt; M.</p> Signup and view all the answers

    What does the Triangle Inequality imply when considering the boundedness of the set A = {z ∈ C | |z - z0| < 1}?

    <p>|z| &lt; 1 + |z0| for all z ∈ A.</p> Signup and view all the answers

    What is the primary characteristic of a sequence?

    <p>It is an infinite list of numbers that can include rational and irrational values.</p> Signup and view all the answers

    Which property must a convergent sequence possess?

    <p>The terms of the sequence must be bounded above.</p> Signup and view all the answers

    What does the proof show about the relationship between boundedness in sequences and the completeness property of the real numbers?

    <p>Every bounded sequence has a limit in the real numbers.</p> Signup and view all the answers

    Study Notes

    Mathematical Logic and Sets

    • Propositions: A statement that is either True or False.

    • Examples of Propositions:

      • 1 + 1 = 2
      • π = 3
      • There exists a real number x such that x2 = x
      • All even integers greater than 2 can be written as the sum of two primes.

    • Predicates: A statement that depends on one or more variables and becomes a proposition when values are assigned to these variables.

    • Examples of Predicates:

      • n is a prime number
      • n+m < 4
      • x2 > y2
      • x2 + 3x + 2 = 0

    Logical Connectives

    • OR: A statement 'P or Q' is True precisely when one or both of P and Q are True.

    • AND: A statement 'P and Q' is True precisely when both P and Q are True.

    • Truth Tables: Used to summarize the truth values of various logical combinations of propositions.

    Implications

    • P implies Q: True in all cases except when P is True and Q is False.
    • Denotation: P)Q

    Sets

    • Definition: A collection of objects.
    • Elements: Objects within a set.
    • Notation: x ∈ A means 'x is an element of A' and x /2 A means 'x is not an element of A'.
    • Finite Sets: Sets with a limited number of elements.
    • Infinite Sets: Sets with an unlimited number of elements.
    • Empty Set: A set with no elements. Denoted by ∅
    • Standard Number Sets:
      • Natural Numbers (N): {1, 2, 3, ...}
      • Integers (Z): {..., -2, -1, 0, 1, 2, ...}
      • Rational Numbers (Q): {a/b | a, b ∈ Z, b ≠ 0}
      • Real Numbers (R): Includes all rational and irrational numbers.
      • Complex Numbers (C): {a + bi | a, b ∈ R}

    Subsets

    • Definition: A set A is a subset of a set B (A ⊆ B) if every element of A is also an element of B.
    • Proper Subset A is a proper subset of B (A ⊂ B) if A is a subset of B and A is not equal to B.

    Operations on Sets

    • Intersection (A ∩ B): The set of elements that are in both A and B.
    • Union (A ∪ B): The set of elements that are in A or B (or both).
    • Difference (A \ B): The set of elements that are in A but not in B.
    • Complement (Ac): The set of elements that are in the universal set U but not in A.

    Quantifiers

    • There exists (∃): Means 'there is at least one'.
    • For all (∀): Means 'for every'.

    Methods of Proof

    • Direct Proof: Starts with assumptions and uses implications to reach a conclusion.
    • Proof by Contradiction: Assumes a statement is false and derives a contradiction to prove the statement is true.

    Proof by Induction

    • Base Case: Proving the statement is true for the first case (often n=1).
    • Inductive Step: Assuming the statement is true for some value k and proving that it is also true for k+1. This proves that the statement holds for all natural numbers n.

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