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Unit 6 Review and Checkpoint

A free Logic lesson from the “Biconditionals and Definitions” unit, with a worked example and practice problems including step-by-step solutions.

This checkpoint checks whether learners can use two-way reasoning without confusing it with a one-way conditional. Learning objective: Review iff statements, definitions, examples, and necessary/sufficient 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

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 6 Review and Checkpoint): What does p ↔ q mean?

  1. Checkpoint Practice: First identify exactly what the question is asking: Example case A (Unit 6 Review and Checkpoint): What does p ↔ q mean?
  2. For data questions, identify what each statistic measures before calculating so the result matches the question.
  3. A biconditional is two-way.
  4. It contains both conditional directions.

Answer: p implies q and q implies p

Practice problems

1. Practice case A (Unit 6 Review and Checkpoint): What does p ↔ q mean?

Choices: p implies q and q implies p · p implies q only · p and q are both false · not p or q

Show solution
  1. Checkpoint Practice: First identify exactly what the question is asking: Practice case A (Unit 6 Review and Checkpoint): What does p ↔ q mean?
  2. For data questions, identify what each statistic measures before calculating so the result matches the question.
  3. A biconditional is two-way.
  4. It contains both conditional directions.
  5. So p ↔ q means each statement implies the other.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: p implies q and q implies p

2. Practice case B (Unit 6 Review and Checkpoint): In "If a figure is a square, then the figure is a rectangle," what is the conclusion?

Choices: the figure is a rectangle · a figure is a square · if · only if

Show solution
  1. The conclusion is the then-part.
  2. It is what follows if the hypothesis holds.
  3. Here the conclusion is the figure is a rectangle.

Answer: the figure is a rectangle

3. Practice case C (Unit 6 Review and Checkpoint): A definition is too broad when it:

Choices: includes things that should be excluded · excludes correct examples · uses symbols · has examples

Show solution
  1. Too broad means the category catches extra objects.
  2. For example, defining square as any four-sided figure includes rectangles that are not squares.
  3. So it includes too much.

Answer: includes things that should be excluded

4. Practice case D (Unit 6 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 6 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 6 Review and Checkpoint): Which object is a non-example for "a square is a rectangle with four equal sides"?

Choices: A 3 by 5 rectangle · A 4 by 4 square · A square tile · A rectangle with all sides equal

Show solution
  1. Checkpoint Practice: First identify exactly what the question is asking: Practice case E (Unit 6 Review and Checkpoint): Which object is a non-example for "a square is a rectangle with four equal sides"?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. A non-example should fail the definition.
  4. A 3 by 5 rectangle does not have four equal sides.
  5. So it is not a square.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: A 3 by 5 rectangle

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