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.
This checkpoint checks whether learners can write and evaluate proof-ready reasoning before moving into formal proof-heavy courses. Learning objective: Review direct reasoning, counterexamples, cases, contradiction, and explanation critique. Prerequisite: Review the lessons in this unit before starting.. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: A truth-table question asks for cases; a counterexample question asks for one case that breaks a claim. Example 2: A validity question asks whether the conclusion must follow, not whether the sentences sound realistic. A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.
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. Example case A (Unit 11 Review and Checkpoint): Worked example: Which rule or habit best matches Unit 11 Review and Checkpoint?
- Unit 11 Review and Checkpoint targets a specific reasoning habit.
- Unit 11 Review and Checkpoint focuses on review direct reasoning, counterexamples, cases, contradiction, and explanation critique.
- The other choices either overclaim or change the logical relationship.
Answer: Unit 11 Review and Checkpoint focuses on review direct reasoning, counterexamples, cases, contradiction, and explanation critique.
Practice problems
1. Practice case A (Unit 11 Review and Checkpoint): Which explanation is most complete?
Choices: Because x is even, x = 2k for an integer k, so x + 2 = 2(k + 1), which is even. · It stays even. · I tried x = 4. · The answer looks right.
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case A (Unit 11 Review and Checkpoint): Which explanation is most complete?
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- A complete explanation uses a definition.
- It shows the algebraic step.
- It connects the result back to evenness.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: Because x is even, x = 2k for an integer k, so x + 2 = 2(k + 1), which is even.
2. Practice case B (Unit 11 Review and Checkpoint): Name the form: If p then q. Not q. Therefore not p.
Choices: Modus tollens · Modus ponens · Affirming the consequent · Hypothetical syllogism
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case B (Unit 11 Review and Checkpoint): Name the form: If p then q. Not q. Therefore not p.
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- The argument uses p → q.
- It denies q.
- Therefore ¬p follows by modus tollens.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: Modus tollens
3. Practice case C (Unit 11 Review and Checkpoint): "Some A are B" means:
Choices: At least one object is in both A and B · Every A is B · No A is B · Every B is A
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case C (Unit 11 Review and Checkpoint): "Some A are B" means:
- For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
- Some means at least one.
- For set diagrams, that object lies in the overlap.
- It does not mean all.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: At least one object is in both A and B
4. Practice case D (Unit 11 Review and Checkpoint): Reasoning by cases is appropriate when:
Choices: the cases cover all possibilities · only one example is checked · the conclusion is ignored · the domain is unknown
Show solution
- Casework splits a problem into possibilities.
- The proof is complete only if every possibility is covered.
- Then each case can be handled separately.
Answer: the cases cover all possibilities
5. Practice case E (Unit 11 Review and Checkpoint): Which form is invalid? If p then q. q. Therefore p.
Choices: Affirming the consequent · Modus ponens · Modus tollens · Disjunctive syllogism
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case E (Unit 11 Review and Checkpoint): Which form is invalid? If p then q. q. Therefore p.
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- The conclusion may have another cause.
- q being true does not force p.
- So this is affirming the consequent.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: Affirming the consequent
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