Problem 1
There exists x such that for every y, not P(x,y).
Faculty solution guide
Predicates, Domains and Quantifier Scope
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Predicates, Domains and Quantifier Scope.
Problem 4
The argument must address translating open sentences into precisely quantified statements with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in formal verification, database queries, proof assistants, specification languages, and research hypotheses, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.