Unit 5 Review and Checkpoint
A free Logic lesson from the “Conditionals” unit, with a worked example and practice problems including step-by-step solutions.
An implication p→q rules out p true with q false. Its converse q→p need not follow. Its contrapositive ¬q→¬p is equivalent. In a concrete application identify which condition is actually given before applying a theorem. This review checks performance on mathematical tasks. For extended written work, use the separate rubric with a tutor or teacher; the automatic quiz does not certify proof-writing proficiency.
What you'll learn
- Review conditional statements, converses, inverses, contrapositives, and necessary/sufficient language
- Choose the reasoning tool that matches the statement
- Explain why an answer is valid, invalid, true, false, or unsupported
Worked example
Problem. Complete the final column for p → q. Row order (p,q): TT, TF, FT, FF.
- At T, T, p → q is true.
- At T, F, p → q is false.
- At F, T, p → q is true.
- At F, F, p → q is true.
Answer: T, F, T, T
Lesson preview: this page shows 5 practice items. The interactive activity contains 8 practice items and 8 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Complete the final column for p → q. Row order (p,q): TT, TF, FT, FF.
Choices: F, T, F, F · F, F, T, T · T, F, T, F · T, F, T, T
Show solution
- At T, T, p → q is true.
- At T, F, p → q is false.
- At F, T, p → q is true.
- At F, F, p → q is true.
Answer: T, F, T, T
2. If an integer is divisible by 14, it is even. Which fact alone is sufficient to use this implication?
Choices: The integer is not divisible by the given number · The integer is positive · The integer is divisible by 14 · The integer is even
Show solution
- The antecedent triggers the implication.
- The consequent alone does not establish the antecedent.
Answer: The integer is divisible by 14
3. The rule is “If a key is valid, the door opens.” Which observation disproves the rule?
Choices: A valid key is used and the door opens · A valid key is used and the door does not open · An invalid key is used and the door opens · An invalid key is used and the door stays shut
Show solution
- An implication fails exactly when its hypothesis is true and its conclusion false.
Answer: A valid key is used and the door does not open
4. For p→q, which statement is its converse?
Choices: q → p · ¬p → ¬q · ¬q → ¬p · p ∧ q
Show solution
- The converse swaps hypothesis and conclusion without negating them.
Answer: q → p
5. A workshop rule says “Admitted implies registered.” Which is necessary for admission?
Choices: Having a blue badge · Being registered guarantees admission · Being unregistered · Being registered
Show solution
- Necessary means it must hold whenever admission occurs.
- The rule requires registration but does not promise admission to every registered person.
Answer: Being registered
Practice this interactively with instant feedback and an AI tutor.
Practice Unit 5 Review and Checkpoint Take the free placement check