Unit 7 Review and Checkpoint
A free Logic lesson from the “Logical Equivalence” unit, with a worked example and practice problems including step-by-step solutions.
This checkpoint checks whether learners can preserve meaning while rewriting statements. Learning objective: Review equivalence, De Morgan's Laws, contrapositives, and truth-table tests. 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 equivalence, De Morgan's Laws, contrapositives, and truth-table tests
- 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 7 Review and Checkpoint): Which statement is equivalent to ¬(p ∧ q)?
- Checkpoint Practice: First identify exactly what the question is asking: Example case A (Unit 7 Review and Checkpoint): Which statement is equivalent to ¬(p ∧ q)?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Use De Morgan's Law.
- Negating an and changes it to or.
Answer: ¬p ∨ ¬q
Practice problems
1. Practice case A (Unit 7 Review and Checkpoint): Which statement is equivalent to ¬(p ∧ q)?
Choices: ¬p ∨ ¬q · ¬p ∧ ¬q · p ∨ q · p ∧ ¬q
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case A (Unit 7 Review and Checkpoint): Which statement is equivalent to ¬(p ∧ q)?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Use De Morgan's Law.
- Negating an and changes it to or.
- Negate both parts: ¬p ∨ ¬q.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: ¬p ∨ ¬q
2. Practice case B (Unit 7 Review and Checkpoint): A truth table with two variables has how many rows?
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case B (Unit 7 Review and Checkpoint): A truth table with two variables has how many rows?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Each variable has two truth values.
- Two variables create 2 x 2 cases.
- That gives 4 rows.
- Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.
Answer: 4
3. Practice case C (Unit 7 Review and Checkpoint): What is the converse of "If a student scores at least 70, then the quiz is passed"?
Choices: If the quiz is passed, then a student scores at least 70. · If not a student scores at least 70, then not the quiz is passed. · If not the quiz is passed, then not a student scores at least 70. · a student scores at least 70 and the quiz is passed.
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case C (Unit 7 Review and Checkpoint): What is the converse of "If a student scores at least 70, then the quiz is passed"?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The converse switches the hypothesis and conclusion.
- It does not negate them.
- So q -> p is the converse.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: If the quiz is passed, then a student scores at least 70.
4. Practice case D (Unit 7 Review and Checkpoint): How can a truth table show two statements are equivalent?
Choices: Their final columns match in every row. · They use the same number of letters. · One statement is longer. · Both contain an arrow.
Show solution
- Equivalence means same truth value in every case.
- Truth tables list every case.
- Matching final columns prove equivalence.
Answer: Their final columns match in every row.
5. Practice case E (Unit 7 Review and Checkpoint): In the row p=False, q=True, r=True, what is p ↔ q?
Choices: True · False
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case E (Unit 7 Review and Checkpoint): In the row p=False, q=True, r=True, what is p ↔ q?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- p is False and q is True.
- A biconditional is true when both parts have the same truth value.
- The final value is False.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: False
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