This checkpoint draws from every instructional section in this chapter. It is not a new topic or a substitute for individual sections. Explain your method before selecting an answer, and use paper or the scratch space for intermediate work. A correct guess is useful diagnostic information but not secure knowledge.

After an error, identify whether it came from notation, selecting a rule, missing a condition, algebra/arithmetic, or execution. Open the linked repair lesson, solve a different problem, then return to mixed practice. Reading the solution marks assistance, not independent success. Your chapter progress depends on actual skill evidence, not on opening this page.

Reconstruct these ideas without notes

Propositions and logical operations: Without looking: when are AND, inclusive OR, and XOR true? Give an open sentence and explain what would turn it into a proposition.
Evaluating compound propositions: Build the input rows for p,q,r from memory. What output pattern distinguishes a tautology from a satisfiable contingency?
Conditional statements: Translate “A is necessary for B.” Write all three related forms of p→q and identify the equivalent one.
Logical equivalence: What is the relationship between P≡Q and P↔Q? How many counterexamples are required to disprove equivalence?
Laws of propositional logic: Simplify p∨(¬p∧q) and name each step. Explain why distribution is useful in this direction.
Predicates and quantifiers: For a predicate over integers, what evidence proves ∃xP(x)? What evidence disproves ∀xP(x)?
Translating quantified statements: Why is ∃x(A(x)→B(x)) not a reliable translation of “some A is B”?
Negating quantified statements: Negate ∀x∈ℝ(P(x)→x<5), with no negation left in front of a quantifier or implication.
Nested quantifiers: Explain the difference between ∀x∃y and ∃y∀x without using the word “order” alone.
Logical reasoning and validity: What exact truth-value pattern makes an argument invalid? How is validity different from soundness?
Propositional rules of inference: Reconstruct MP, MT, HS, and DS from memory. Refute affirming the consequent with one assignment.
Quantified rules of inference: Explain why UI may use a witness introduced by EI, but UG cannot turn that witness into “every object.”
Mathematical definitions: Write precise definitions of odd, rational, and a|b. Which witnesses must be integers?
Introduction to proofs: How would the evidence differ for proving and disproving “every integer has property P”?
Proof quality and common errors: Give an example of a true statement with an invalid proof. Explain the first invalid step rather than just saying “wrong.”
Writing direct proofs: Prove that a multiple of 6 is even. Identify the witness at the end. Then outline a proof that rational numbers are closed under multiplication.
Proof by contrapositive: Form the contrapositive of “if x>0 and y>0, then xy>0.” What connective appears in its conclusion?
Proof by contradiction: Reconstruct the √2 proof and explain the role of lowest terms. How does its setup differ from a contrapositive proof?
Proof by cases: Prove n(n+1) is even in two cases. Why would checking n=1 and n=2 not be the same proof?