These seven questions use the skills in proving or disproving an everyday mathematical claim. They ramp from a quick counterexample to a general argument.
Write a full answer before opening each block.
Questions
Question 1. Claim: “For every positive whole number n, n² is greater than n.” Find a counterexample.
Answer
Try n = 1. Then n² = 1, which is equal to n, not greater than n. The claim fails, so n = 1 is a counterexample.
Question 2. Claim: “The sum of any two prime numbers is even.” Disprove it.
Answer
Take 2 and 3, which are both prime. Their sum is 5, which is odd. One pair is enough, so the claim is false.
Question 3. Claim: “For every whole number n from 1 upward, 2n + 1 is prime.” Find the smallest counterexample.
Answer
Test in order: n = 1 gives 3, n = 2 gives 5, n = 3 gives 7, and n = 4 gives 9. Since 9 = 3 × 3, it is not prime.
The smallest counterexample is n = 4.
Question 4. Premise 1: All multiples of 5 end in 0 or 5. Premise 2: 35 ends in 5.
Conclusion: 35 is a multiple of 5. Is the argument valid? Change the property to show the pattern of reasoning can fail.
Answer
The argument is invalid in form, because it reverses the first premise. Here the conclusion is true only because every number ending in 0 or 5 happens to be a multiple of 5.
Change the property and see the pattern fail: “All multiples of 10 are divisible by 5. 25 is divisible by 5. So 25 is a multiple of 10.” The premises are true and the conclusion is false.
Question 5. Let A = {2, 4, 6} and B = {1, 2, 3, 4, 5, 6}. Write the implication from A ⊂ B, the converse, and a counterexample to the converse.
Answer
Implication: if x is in A, then x is in B. This is true because every element of A is listed in B.
Converse: if x is in B, then x is in A. This is false, because 1 is in B and not in A.
Question 6. The number of regions when you join points on a circle is 1, 2, 4, 8, 16 for one to five points. A student says six points give 32. Explain what the pattern does and does not show.
Answer
The pattern suggests doubling, but five cases do not prove it continues. Counting regions for six points gives 31, not 32, so the doubling pattern fails.
The right wording is: “The first five cases suggest doubling, but this has not been proved.”
Question 7. Show that the product of two odd numbers is always odd.
Answer
Write the two odd numbers as 2a + 1 and 2b + 1, where a and b are whole numbers.
Multiply: (2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1.
This has the form 2 × (a whole number) + 1, so it is odd for every choice of a and b. Examples such as 3 × 5 = 15 support the result, and the algebra proves it.
If you got these wrong
- Questions 1 to 3: read finding one counterexample.
- Question 4: read separating a valid argument from a true conclusion.
- Question 5: read building an implication from a pair of set relationships.
- Question 6: read when a diagram suggests a result but does not establish it.
The timed original practice session builder lets you repeat the set under time, and the mistake log keeps track of repeats. For a teacher who can watch you test claims, see online one-to-one Mathematics tuition.