Podcast
Questions and Answers
What concept is tested in TMUA 2021 related to integers?
What concept is tested in TMUA 2021 related to integers?
Which TMUA exam year features prime numbers as a topic?
Which TMUA exam year features prime numbers as a topic?
Which of the following was a focus in both TMUA 2021 assessments?
Which of the following was a focus in both TMUA 2021 assessments?
What type of statements are examined in TMUA 2023 regarding mathematical expressions?
What type of statements are examined in TMUA 2023 regarding mathematical expressions?
Signup and view all the answers
Which content area was deemed medium difficulty across two different TMUA years?
Which content area was deemed medium difficulty across two different TMUA years?
Signup and view all the answers
Study Notes
Counterexamples: Statements with Exponents
- Claim: For all positive real numbers x and y, √x = x√y
- Counterexample I: x = 1, y = 16. √1 = 1 and 1√16 = 1*4 = 4. This is not a counterexample as it does not satisfy the claim.
- Counterexample II: x = 2, y = 8. √2 ≠ 2√8 = 2*2√2. This is a counterexample.
- Counterexample III: x = 3, y = 4. √3 ≠ 3√4 = 3*2=6. This is not a counterexample.
- Correct Answer: II only
Counterexamples: Prime Numbers
- Claim: If n is prime, then n² + 2 is not prime.
- Counterexample I: n = 2. 2² + 2 = 6, which is not prime. This is not a counterexample, as the claim is satisfied.
- Counterexample II: n = 3. 3² + 2 = 11, which is prime. This is a counterexample.
- Counterexample III: n = 4. 4² + 2 = 18. This is not prime. Not a counterexample.
- Correct Answer: II only
Counterexamples: Factors of Integers
- Claim: If a is a factor of bc, then a is a factor of b or a is a factor of c.
- Counterexample I: a = 5, b = 10, c = 20. 5 is a factor of 10*20, and 5 is a factor of 10, so this is not a counterexample.
- Counterexample II: a = 8, b = 4, c = 4. 8 is a factor of 4*4. 8 is not a factor of 4. This is a counterexample.
- Counterexample III: a = 6, b = 7, c = 12. 6 is a factor of 7*12 = 84, and 6 is a factor of 12, so this is not a counterexample.
- Correct Answer: II only
Counterexamples: Sequence of Positive Integers
- Sequence: u₁ = 15, u₂ = 21, u₃ = 30, u₄ = 37, u₅ = 44, u₆ = 51, u₇ = 59
- Claim: If n is a prime number, then uₙ is a multiple of 3 or uₙ is a multiple of 5.
- Counterexample: n = 5. u₅ = 44 is not a multiple of 3 or 5. This is the smallest counterexample.
- Correct Answer: 5
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Related Documents
Description
Test your understanding of mathematical claims and counterexamples in this quiz. Analyze different statements involving exponents, prime numbers, and factors of integers. Can you identify the correct counterexamples that disprove the claims?
