Unit 9 Review and Checkpoint
A free Logic lesson from the “Quantifiers and Predicates” unit, with a worked example and practice problems including step-by-step solutions.
Translate all A are B as ∀x(A(x)→B(x)) and some A are B as ∃x(A(x)∧B(x)). Negation switches every to at least one and vice versa while negating the inner claim. Quantifier order matters: ∀x∃y permits y to depend on x; ∃y∀x requires one fixed y to work for every x. 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 predicates, domains, all/some claims, and quantified negation
- Choose the reasoning tool that matches the statement
- Explain why an answer is valid, invalid, true, false, or unsupported
Worked example
Problem. Let S(x) mean student and R(x) mean registered, with all people as the domain. Translate “Every student is registered.”
- Only students must satisfy registration.
- An implication expresses that restriction without saying everyone is a student.
Answer: ∀x(S(x) → R(x))
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. For P(x): x > 7, evaluate P(8).
Choices: True · False
Show solution
- 8 exceeds 7.
Answer: True
2. For P(x): x > 7, evaluate P(7).
Choices: True · False
Show solution
- Strict inequality excludes equality.
Answer: False
3. For domain {7, 8, 9} and P(x): x is even, which values satisfy P?
Choices: All three values · No values · 8 · 7, 9
Show solution
- Test the stated predicate separately at every domain value.
Answer: 8
4. Over positive integers, ∃x(x²=2) is false. What happens if the domain changes to positive real numbers?
Choices: Every real number becomes a witness · It becomes true because √2 is allowed · It stays false because 2 is not a perfect square · It becomes a command
Show solution
- Existential truth depends on whether the domain contains a witness.
Answer: It becomes true because √2 is allowed
5. Let S(x) mean student and R(x) mean registered, with all people as the domain. Translate “Every student is registered.”
Choices: ∀x(S(x) → R(x)) · ∀x(S(x) ∧ R(x)) · ∃x(S(x) → R(x)) · ∀x(R(x) → S(x))
Show solution
- Only students must satisfy registration.
- An implication expresses that restriction without saying everyone is a student.
Answer: ∀x(S(x) → R(x))
Practice this interactively with instant feedback and an AI tutor.
Practice Unit 9 Review and Checkpoint Take the free placement check