Unit 10 Review and Checkpoint
A free Logic lesson from the “Counterexamples, Sets, and Diagrams” unit, with a worked example and practice problems including step-by-step solutions.
This checkpoint checks whether learners can use examples, diagrams, and set relationships to test claims. Learning objective: Review counterexamples, sets, diagrams, and classification reasoning. 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 counterexamples, sets, diagrams, and classification reasoning
- 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 10 Review and Checkpoint): Worked example: Which rule or habit best matches Unit 10 Review and Checkpoint?
- Unit 10 Review and Checkpoint targets a specific reasoning habit.
- Unit 10 Review and Checkpoint focuses on review counterexamples, sets, diagrams, and classification reasoning.
- The other choices either overclaim or change the logical relationship.
Answer: Unit 10 Review and Checkpoint focuses on review counterexamples, sets, diagrams, and classification reasoning.
Practice problems
1. Practice case A (Unit 10 Review and Checkpoint): Which number is a counterexample to "All odd numbers are prime"?
Choices: 9 · 3 · 5 · 7
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case A (Unit 10 Review and Checkpoint): Which number is a counterexample to "All odd numbers are prime"?
- For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
- A counterexample must be odd and not prime.
- 9 is odd.
- 9 is not prime, so it disproves the claim.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: 9
2. Practice case B (Unit 10 Review and Checkpoint): Which phrase signals an existential statement?
Choices: at least one · every · all · no exceptions
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case B (Unit 10 Review and Checkpoint): Which phrase signals an existential statement?
- For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
- Existential statements claim an example exists.
- At least one means some object has the property.
- That is existential.
- 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
3. Practice case C (Unit 10 Review and Checkpoint): Name the form: If p then q. If q then r. Therefore if p then r.
Choices: Hypothetical syllogism · Disjunctive syllogism · Denying the antecedent · Biconditional definition
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case C (Unit 10 Review and Checkpoint): Name the form: If p then q. If q then r. Therefore if p then r.
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- The argument chains conditionals.
- p leads to q, and q leads to r.
- So p leads to r.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: Hypothetical syllogism
4. Practice case D (Unit 10 Review and Checkpoint): "No A are B" means:
Choices: The sets do not overlap · A is inside B · B is inside A · At least one object is in both
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case D (Unit 10 Review and Checkpoint): "No A are B" means:
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- No A are B rules out shared members.
- A Venn diagram would show disjoint sets.
- So there is no overlap.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: The sets do not overlap
5. Practice case E (Unit 10 Review and Checkpoint): In "For every x in the integers, x + 0 = x," what is the domain?
Choices: the integers · x + 0 · x · 0
Show solution
- The domain names the objects being discussed.
- The phrase in the integers gives the domain.
- So the domain is the integers.
Answer: the integers
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