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Mixed Logic Review

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

Mixed review asks students to decide which tool fits: truth table, negation, counterexample, quantifier, diagram, or argument form. Learning objective: Choose among logic tools across the full course. Prerequisite: No formal prerequisite. 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 (Mixed Logic Review): Classify the sentence "The number 18 is divisible by 3."

  1. It makes a claim that can be checked.
  2. Logic starts by deciding whether a sentence makes a true-or-false claim.
  3. The best classification is Statement.

Answer: Statement

Practice problems

1. Practice case A (Mixed Logic Review): Classify the sentence "The number 18 is divisible by 3."

Choices: Statement · Question · Command · Fragment

Show solution
  1. It makes a claim that can be checked.
  2. Logic starts by deciding whether a sentence makes a true-or-false claim.
  3. The best classification is Statement.

Answer: Statement

2. Practice case B (Mixed Logic Review): Which is a compound statement?

Choices: x is positive and x is even. · x is positive. · What is x? · Solve for x.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case B (Mixed Logic Review): Which is a compound statement?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. A compound statement joins simpler claims.
  4. The word and connects two claims.
  5. So the first choice is compound.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x is positive and x is even.

3. Practice case C (Mixed Logic Review): Simplify the double negation ¬¬p.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case C (Mixed Logic Review): Simplify the double negation ¬¬p.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. The first negation flips p.
  4. The second negation flips it back.
  5. So ¬¬p is equivalent to p.
  6. Check the result by substituting or estimating: the response should match p and make sense in the original problem.

Answer: p

4. Practice case D (Mixed Logic Review): Which symbolic form matches "p or q" in standard mathematical logic?

Choices: p ∨ q · p ∧ q · ¬p · p ↔ q

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case D (Mixed Logic Review): Which symbolic form matches "p or q" in standard mathematical logic?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The word or is represented by ∨.
  4. Standard mathematical or is inclusive.
  5. So p or q becomes p ∨ q.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: p ∨ q

5. Practice case E (Mixed Logic Review): In the row p=False, q=True, r=True, what is p ↔ q?

Choices: True · False

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice case E (Mixed Logic Review): 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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