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Unit 2 Review and Checkpoint

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

This checkpoint checks whether learners can identify claims and form exact opposites in both words and symbols. Learning objective: Review simple statements, compound statements, and precise negation. 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 2 Review and Checkpoint): Worked example: Which rule or habit best matches Unit 2 Review and Checkpoint?

  1. Unit 2 Review and Checkpoint targets a specific reasoning habit.
  2. Unit 2 Review and Checkpoint focuses on review simple statements, compound statements, and precise negation.
  3. The other choices either overclaim or change the logical relationship.

Answer: Unit 2 Review and Checkpoint focuses on review simple statements, compound statements, and precise negation.

Practice problems

1. Practice case A (Unit 2 Review and Checkpoint): Is "Every square has four equal sides." a logical statement?

Choices: Yes, it makes a claim. · No, it is only a command. · No, it is only a question. · No, statements cannot use numbers.

Show solution
  1. Checkpoint Practice: First identify exactly what the question is asking: Practice case A (Unit 2 Review and Checkpoint): Is "Every square has four equal sides." a logical statement?
  2. Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  3. It makes a mathematical claim.
  4. A logical statement needs a possible truth value.
  5. This sentence qualifies.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: Yes, it makes a claim.

2. Practice case B (Unit 2 Review and Checkpoint): Which mistake is common when negating "All dogs bark"?

Choices: Writing 'No dogs bark' instead of 'At least one dog does not bark' · Changing all to every · Keeping the same topic · Looking for a counterexample

Show solution
  1. Checkpoint Practice: First identify exactly what the question is asking: Practice case B (Unit 2 Review and Checkpoint): Which mistake is common when negating "All dogs bark"?
  2. For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
  3. The opposite of all is not none.
  4. To make all false, one counterexample is enough.
  5. No dogs bark is stronger than needed.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: Writing 'No dogs bark' instead of 'At least one dog does not bark'

3. Practice case C (Unit 2 Review and Checkpoint): If p means "the number is even," write the symbolic form of "not p."

Show solution
  1. Checkpoint Practice: First identify exactly what the question is asking: Practice case C (Unit 2 Review and Checkpoint): If p means "the number is even," write the symbolic form of "not p."
  2. Read the statement in plain English first, then match the symbol, connective, quantifier, or argument form to that meaning.
  3. The symbol ¬ means not.
  4. Place ¬ before the statement letter.
  5. The symbolic form is ¬p.
  6. Check the response against the original logical form: it should match ¬p without changing the meaning.

Answer: ¬p

4. Practice case D (Unit 2 Review and Checkpoint): The negation of "The answer is at least 12" is:

Choices: The answer is less than 12 · The answer is greater than 12 · The answer is exactly 12 · The answer is at most 12

Show solution
  1. Checkpoint Practice: First identify exactly what the question is asking: Practice case D (Unit 2 Review and Checkpoint): The negation of "The answer is at least 12" is:
  2. Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  3. At least 12 means 12 or more.
  4. The opposite is anything below 12.
  5. So the answer is less than 12.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: The answer is less than 12

5. Practice case E (Unit 2 Review and Checkpoint): Which phrase usually signals a conclusion?

Choices: therefore · because · given that · assume

Show solution
  1. Therefore points to what follows.
  2. Because and given that often introduce reasons.
  3. So therefore is the conclusion signal.

Answer: therefore

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