Unit 5 Review and Checkpoint
A free Logic lesson from the “Conditionals” unit, with a worked example and practice problems including step-by-step solutions.
This checkpoint checks if-then reasoning before learners use it heavily in definitions, proofs, and theorems. Learning objective: Review conditional statements, converses, inverses, contrapositives, and necessary/sufficient language. 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 conditional statements, converses, inverses, contrapositives, and necessary/sufficient language
- 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 5 Review and Checkpoint): Worked example: Which rule or habit best matches Unit 5 Review and Checkpoint?
- Unit 5 Review and Checkpoint targets a specific reasoning habit.
- Unit 5 Review and Checkpoint focuses on review conditional statements, converses, inverses, contrapositives, and necessary/sufficient language.
- The other choices either overclaim or change the logical relationship.
Answer: Unit 5 Review and Checkpoint focuses on review conditional statements, converses, inverses, contrapositives, and necessary/sufficient language.
Practice problems
1. Practice case A (Unit 5 Review and Checkpoint): In "If a number is divisible by 4, then the number is even," what is the hypothesis?
Choices: a number is divisible by 4 · the number is even · if · then
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case A (Unit 5 Review and Checkpoint): In "If a number is divisible by 4, then the number is even," what is the hypothesis?
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- The hypothesis is the if-part.
- The conclusion is the then-part.
- Here the hypothesis is a number is divisible by 4.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: a number is divisible by 4
2. Practice case B (Unit 5 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 5 Review and Checkpoint): A truth table with two variables has how many rows?
- For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
- Each variable has two truth values.
- Two variables create 2 x 2 cases.
- That gives 4 rows.
- Check the response against the original logical form: it should match 4 without changing the meaning.
Answer: 4
3. Practice case C (Unit 5 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 5 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 5 Review and Checkpoint): What is the contrapositive of "If a triangle is equilateral, then the triangle is isosceles"?
Choices: If the triangle is not isosceles, then a triangle is not equilateral. · If the triangle is isosceles, then a triangle is equilateral. · If a triangle is not equilateral, then the triangle is not isosceles. · If a triangle is equilateral, then the triangle is not isosceles.
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case D (Unit 5 Review and Checkpoint): What is the contrapositive of "If a triangle is equilateral, then the triangle is isosceles"?
- For a contrapositive, switch the hypothesis and conclusion, then negate both parts.
- The contrapositive switches and negates both parts.
- p -> q becomes ¬q -> ¬p.
- That is the first choice.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: If the triangle is not isosceles, then a triangle is not equilateral.
5. Practice case E (Unit 5 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 5 Review and Checkpoint): In the row p=False, q=True, r=True, what is p ↔ q?
- For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
- 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 preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: False
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