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
- Choose among logic tools across the full course
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Example case A (Mixed Logic Review): Worked example: Which rule or habit best matches Mixed Logic Review?
- Mixed Logic Review targets a specific reasoning habit.
- Mixed Logic Review focuses on choose among logic tools across the full course.
- The other choices either overclaim or change the logical relationship.
Answer: Mixed Logic Review focuses on choose among logic tools across the full course.
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
- It makes a claim that can be checked.
- Logic starts by deciding whether a sentence makes a true-or-false claim.
- 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
- Warm-up: First identify exactly what the question is asking: Practice case B (Mixed Logic Review): Which is a compound statement?
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- A compound statement joins simpler claims.
- The word and connects two claims.
- So the first choice is compound.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: x is positive and x is even.
3. Practice case C (Mixed Logic Review): Simplify the double negation ¬¬p.
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case C (Mixed Logic Review): Simplify the double negation ¬¬p.
- Read the statement in plain English first, then match the symbol, connective, quantifier, or argument form to that meaning.
- The first negation flips p.
- The second negation flips it back.
- So ¬¬p is equivalent to p.
- Check the response against the original logical form: it should match p without changing the meaning.
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
- 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?
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- The word or is represented by ∨.
- Standard mathematical or is inclusive.
- So p or q becomes p ∨ q.
- 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 ∨ q
5. Practice case E (Mixed Logic Review): Lesson focus: Which transfer task would show readiness for Mixed Logic Review?
Choices: Apply choose among logic tools across the full course in a new sentence or context. · Repeat the exact worked example without changing the context. · Pick an answer before identifying the claim parts. · Use a rule from another lesson because the words sound close.
Show solution
- A transfer task keeps the same objective but changes the surface context.
- Apply choose among logic tools across the full course in a new sentence or context.
- That is a better mastery signal than memorizing a practice prompt.
Answer: Apply choose among logic tools across the full course in a new sentence or context.
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