Unit 11 Review and Checkpoint
A free Logic lesson from the “Proof Readiness” unit, with a worked example and practice problems including step-by-step solutions.
Begin with arbitrary allowed objects and their definitions. Use justified algebra to obtain the target form, then explain why it proves the claim. For parity, specify that the multiplier is an integer. A numerical example illustrates a proof but cannot replace the arbitrary-object argument. This review checks performance on mathematical tasks. For extended written work, use the separate rubric with a tutor or teacher; the automatic quiz does not certify proof-writing proficiency.
What you'll learn
- Review direct reasoning, counterexamples, cases, contradiction, and explanation critique
- Choose the reasoning tool that matches the statement
- Explain why an answer is valid, invalid, true, false, or unsupported
Worked example
Problem. Let a=2r and b=2s for integers r,s. Which line proves a+b is even?
- Substitute the definitions and factor 2.
- Closure of integers under addition supplies an integer multiplier.
Answer: a+b=2(r+s), and r+s is an integer
Lesson preview: this page shows 5 practice items. The interactive activity contains 16 practice items and 16 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Let a=2r and b=2s for integers r,s. Which line proves a+b is even?
Choices: a+b=2(r+s), and r+s is an integer · a+b=r+s · a+b is even because it looks even · Take a=2 and b=4 only
Show solution
- Substitute the definitions and factor 2.
- Closure of integers under addition supplies an integer multiplier.
Answer: a+b=2(r+s), and r+s is an integer
2. For odd integers a=2r+1 and b=2s+1, simplify a+b into even form.
Choices: 2(r+s)+1 · 2rs+1 · r+s+2 · 2(r+s+1)
Show solution
- Add the two expressions: 2r+2s+2.
- Factor out 2.
Answer: 2(r+s+1)
3. To prove the square of any even integer is even, write n=2k. Which next calculation establishes the result?
Choices: n²=2k · Check only n=4 · n²=4k²=2(2k²) · n²=2k²+1
Show solution
- Square both factors, then display an integer multiple of 2.
Answer: n²=4k²=2(2k²)
4. A proof says n=2k, then n+6=2(k+3). Which missing fact completes its parity conclusion?
Choices: Every real number is even · k+3 is an integer · k must be positive · n must equal 6
Show solution
- The definition of even requires an integer multiplier, not just a factor of 2.
Answer: k+3 is an integer
5. Which integer disproves “Every multiple of 10 is a multiple of 20”?
Choices: 10 · 20 · 40 · 60
Show solution
- 10 satisfies the hypothesis but is not divisible by 20.
Answer: 10
Explain and apply
Complete these tasks on paper. Use the rubric to review your reasoning with a teacher or study partner; these tasks are not automatically scored.
Write a direct proof
Prove that the sum of an even integer and an odd integer is odd. Write a proof for arbitrary integers rather than a list of examples.
Review rubric and solution
- Represent the integers as 2r and 2s+1 with r,s integers.
- Derive 2(r+s)+1.
- State that r+s is an integer and invoke the definition of odd.
- Let a=2r and b=2s+1 for integers r,s.
- Then a+b=2r+2s+1=2(r+s)+1.
- Because r+s is an integer, the sum is odd.
Practice this interactively with instant feedback and an AI tutor.
Practice Unit 11 Review and Checkpoint Take the free placement check