Mathematical Proof and Logic Rules Quiz
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Questions and Answers

What is the strategy used in proofs by contradiction?

  • Assume the hypothesis is true and the conclusion is false. (correct)
  • Assume the hypothesis and conclusion are true.
  • Negate both the hypothesis and conclusion.
  • Prove the hypothesis directly.
  • In the proof that n² is even implies n is even, what does assuming n is odd lead to?

  • A direct proof.
  • A counterexample.
  • A contradiction. (correct)
  • An implication.
  • If P → Q is proved by contradiction, what conclusion can be drawn?

  • Q → P is true.
  • P → Q is sometimes true.
  • P → Q is true. (correct)
  • P → Q is false.
  • In the proof that (3n+2) odd implies n is odd, why do we assume n is even?

    <p>To create a contradiction.</p> Signup and view all the answers

    What must be concluded if a proof by contradiction reaches a contradiction?

    <p>The assumption was false.</p> Signup and view all the answers

    Why do proofs by contradiction involve deriving a contradiction?

    <p>To show the assumption must be incorrect.</p> Signup and view all the answers

    Based on the provided text, when is an integer considered even?

    <p>When it is equal to 2k, where k is an integer.</p> Signup and view all the answers

    In the proof by contradiction example, what was the initial assumption made about the existence of a largest integer?

    <p>There is a largest integer.</p> Signup and view all the answers

    What method was used in the example to prove that there is no largest integer?

    <p>Proof by Contradiction</p> Signup and view all the answers

    In the contrapositive statement 'If not Q, then not P', what does 'not P' represent in the original statement 'If P, then Q'?

    <p>The hypothesis</p> Signup and view all the answers

    How did the author conclude that an integer must be odd?

    <p>By contradicting the assumption that the integer was even.</p> Signup and view all the answers

    In notation, how is 'not P' represented?

    <p>~P</p> Signup and view all the answers

    What is the definition of an even integer according to the text?

    <p>An integer n is even if there is some integer k such that n = 2k.</p> Signup and view all the answers

    Which of the following statements represents an implication according to the text?

    <p>If P, then Q.</p> Signup and view all the answers

    What is a direct proof according to the text?

    <p>A proof that starts with an initial set of hypotheses and proves the conclusion using simple logical steps.</p> Signup and view all the answers

    Which statement correctly describes integer relationships based on the text?

    <p>No integer is both even and odd.</p> Signup and view all the answers

    What should be the characteristics of the axioms mentioned in the text?

    <p>They should be minimal for simplicity.</p> Signup and view all the answers

    Which type of logical reasoning involves proving a statement by assuming its negation and reaching a contradiction?

    <p>Proof by Contradiction</p> Signup and view all the answers

    Study Notes

    Proofs by Contradiction

    • In proofs by contradiction, a statement's negation is assumed to be true, and then this assumption is shown to lead to a logical contradiction, thereby proving the original statement.
    • The strategy involves deriving a contradiction to conclude the truth of the original statement.

    Examples of Proofs by Contradiction

    • In the proof that n² is even implies n is even, assuming n is odd leads to a contradiction, thereby proving that n must be even.
    • In the proof that (3n+2) odd implies n is odd, n is assumed to be even, and then a contradiction is derived to conclude that n must be odd.

    Implications and Contrapositive Statements

    • If P → Q is proved by contradiction, it can be concluded that if P is true, then Q must also be true.
    • In a contrapositive statement 'If not Q, then not P', 'not P' represents the negation of the original statement 'If P, then Q'.
    • 'Not P' is represented in notation as ~P.

    Properties of Integers

    • An integer is considered even if it can be written in the form 2k, where k is an integer.
    • The definition of an even integer is an integer that can be divided by 2 without leaving a remainder.

    Direct Proofs and Axioms

    • A direct proof involves showing that a statement is true through a series of logical steps, without assuming its negation.
    • Axioms are statements that are assumed to be true without needing to be proven, and should be self-evident, consistent, and non-redundant.

    Logical Reasoning

    • Proving a statement by assuming its negation and reaching a contradiction is a type of logical reasoning called proof by contradiction.
    • In this type of reasoning, the goal is to show that the negation of the statement leads to a logical contradiction, thereby proving the original statement.

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    Description

    This quiz explores the concept of mathematical proofs, including how they connect axioms, definitions, and validated statements to demonstrate the correctness of a statement. It emphasizes the importance of logic rules and agreed-upon axioms, along with the appropriate level of detail for different audiences. Reference book: Thomas Cormen's 'Structure of a Mathematical Proof'.

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