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
- Review iff statements, definitions, examples, and necessary/sufficient 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 6 Review and Checkpoint): Worked example: Which rule or habit best matches Unit 6 Review and Checkpoint?
- Unit 6 Review and Checkpoint targets a specific reasoning habit.
- Unit 6 Review and Checkpoint focuses on review iff statements, definitions, examples, and necessary/sufficient reasoning.
- The other choices either overclaim or change the logical relationship.
Answer: Unit 6 Review and Checkpoint focuses on review iff statements, definitions, examples, and necessary/sufficient reasoning.
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
- Checkpoint Practice: First identify exactly what the question is asking: Practice case A (Unit 6 Review and Checkpoint): What does p ↔ q mean?
- For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
- A biconditional is two-way.
- It contains both conditional directions.
- So p ↔ q means each statement implies the other.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
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
- The conclusion is the then-part.
- It is what follows if the hypothesis holds.
- 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
- Too broad means the category catches extra objects.
- For example, defining square as any four-sided figure includes rectangles that are not squares.
- 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 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 6 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 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
- 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"?
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- A non-example should fail the definition.
- A 3 by 5 rectangle does not have four equal sides.
- So it is not a square.
- 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 3 by 5 rectangle
Practice this interactively with instant feedback and an AI tutor.
Practice Unit 6 Review and Checkpoint Take the free placement check