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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

Why it matters: Mixed review builds the habit of choosing the right reasoning tool for the claim in front of you.

Worked example

Problem. Example case A (Unit 5 Review and Checkpoint): In "If a number is divisible by 4, then the number is even," what is the hypothesis?

  1. Checkpoint Practice: First identify exactly what the question is asking: Example case A (Unit 5 Review and Checkpoint): In "If a number is divisible by 4, then the number is even," what is the hypothesis?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The hypothesis is the if-part.
  4. The conclusion is the then-part.

Answer: a number is divisible by 4

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
  1. 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?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The hypothesis is the if-part.
  4. The conclusion is the then-part.
  5. Here the hypothesis is a number is divisible by 4.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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
  1. 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?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Each variable has two truth values.
  4. Two variables create 2 x 2 cases.
  5. That gives 4 rows.
  6. 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 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
  1. 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.
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The argument chains conditionals.
  4. p leads to q, and q leads to r.
  5. So p leads to r.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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 not the triangle is isosceles, then not a triangle is equilateral. · If the triangle is isosceles, then a triangle is equilateral. · If not a triangle is equilateral, then not the triangle is isosceles. · If a triangle is equilateral, then not the triangle is isosceles.

Show solution
  1. 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"?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. The contrapositive switches and negates both parts.
  4. p -> q becomes ¬q -> ¬p.
  5. That is the first choice.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: If not the triangle is isosceles, then not a triangle is 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
  1. 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?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. p is False and q is True.
  4. A biconditional is true when both parts have the same truth value.
  5. The final value is False.
  6. 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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