Counterexample - Varsity Tutors

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Counterexample

Study Guide

Key Definition

A counterexample is an example that disproves a statement or proposition. For example, if the statement is 'All birds can fly', a penguin is a counterexample.

Important Notes

  • Counterexamples are used in mathematics to disprove statements or theories.
  • A counterexample is often a single instance or example that challenges the truth of a proposition.

Mathematical Notation

Statements are often represented as 'If p, then q'.Remember to use proper notation when solving problems

Why It Works

A counterexample works by disproving a statement. If a statement says something is true for all items in a set, finding one item in that set for which the statement is not true disproves the statement.

Remember

Remember, just one counterexample is enough to disprove a statement!

Quick Reference

Counterexample:An instance which disproves a statement or proposition

Understanding Counterexample

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Beginner Explanation

A counterexample is an example that disproves a statement or proposition. It serves as proof that the statement is not always true.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

Which of the following statements is a counterexample to the statement 'All birds can fly'?

AA sparrow can flyBAn eagle can flyCA penguin cannot flyDA chicken can flyCheck AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

Your friend claims that all teenagers like pop music. How would you use a counterexample to challenge this statement?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Think about a statement often believed to be true. Can you find a counterexample to disprove it?

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

Which of the following is a counterexample to the statement 'All numbers are positive'?

A2B-3C5D10Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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