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How can one example disprove a universal claim?

A counterexample must satisfy the assumptions and contradict the proposed conclusion.

MathematicsTopic · Selected readingSome background helpful
Background and goal

Try the short introduction first, or bring equivalent reading experience. Obtain the specified text before beginning the longer reading.

Test the claim that every increasing function is linear; construct and explain a counterexample.

Useful preparation

Start here if the background is new. Equivalent experience is enough.

Your pathway

5 steps · 14 hours
  1. 01

    Locate the reading

    0.25h · week 1
    Reading notes and task
    First fifteen minutesCheck the edition or article version and locate the passages below. Write the question you want the reading to resolve.
    Your outputEdition and passage record
  2. 02

    Study the opening

    4h · week 1
    College Algebra 2e

    OpenStax · book

    Chapter 3 function examples and Chapter 5 polynomial functions

    Free full text

    Open the reading
    Reading notes and task

    College Algebra 2e. A counterexample must satisfy the assumptions and contradict the proposed conclusion.

    Try thisRead Chapter 3 function examples and Chapter 5 polynomial functions. Reconstruct the central argument using premises and a conclusion.
    Your outputArgument reconstruction
    Check your understanding

    Answer this question in your own words: How can one example disprove a universal claim?

    Hint

    Name the claim, then explain why a reader might accept it.

    Compare with an example answer

    A counterexample must satisfy the assumptions and contradict the proposed conclusion.

    What to look for

    • States a claim that addresses the question
    • Connects the claim to a passage or observation
    • Separates the author’s argument from personal agreement

    Common mistake: Summarising a biography instead of explaining the argument.

    Start this pathway to keep your answers in My learning.

  3. 03

    Read for contrast

    5h · week 2
    College Algebra 2e

    OpenStax · book

    Chapter 3 function examples and Chapter 5 polynomial functions

    Free full text

    Open the reading
    Reading notes and task

    College Algebra 2e. A counterexample must satisfy the assumptions and contradict the proposed conclusion.

    Try thisRead Chapter 3 function examples and Chapter 5 polynomial functions. Compare two passages that put pressure on your initial interpretation.
    Your outputTwo-passage comparison
  4. 04

    Develop an example

    3h · week 3
    Reading notes and task
    Try thisTest the claim that every increasing function is linear; construct and explain a counterexample.
    Your outputWorked example
    Check your understanding

    What would make your activity a convincing example?

    Hint

    Test the claim that every increasing function is linear; construct and explain a counterexample.

    Compare with an example answer

    A useful response carries out the task: Test the claim that every increasing function is linear; construct and explain a counterexample. It records the observation or passage used and explains how it bears on the question.

    What to look for

    • Completes the specific task
    • Shows evidence or reasoning rather than only a conclusion
    • Identifies an alternative explanation or reading

    Common mistake: Presenting an opinion without showing the example it rests on.

    Start this pathway to keep your answers in My learning.

  5. 05

    Write a considered response

    1.75h · week 4
    Reading notes and task
    Try thisWrite a response to “How can one example disprove a universal claim?” that uses both passages and tests its own conclusion. A failed example that violates an assumption is not a counterexample.
    Your outputReasoned response with references
    Check your understanding

    What is one conclusion this reading does not establish?

    Hint

    Test the boundary: A failed example that violates an assumption is not a counterexample.

    Compare with an example answer

    A failed example that violates an assumption is not a counterexample.

    What to look for

    • States a specific limit
    • Explains why the reading does not warrant the stronger conclusion
    • Distinguishes a limitation from dismissing the work

    Common mistake: Treating a useful interpretation as an unrestricted fact.

    Start this pathway to keep your answers in My learning.

All 1 resources

Background, different viewpoints, and further reading.

  • College Algebra 2e

    OpenStaxbookcore

    A counterexample must satisfy the assumptions and contradict the proposed conclusion.

    Chapter 3 function examples and Chapter 5 polynomial functions