(a) Stingy SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer k, find a satisfying assignment in which at most k variables are true, if such an assignment exists. Prove that stingy SAT is NP-hard. (b) The Double SAT problem asks whether a given satisfiability problem has at least two different satisfying assignments. For example, the problem {{V1, V2}, {V1, V2}, {V1, V2}} is satisfiable, but has only one solution (v₁ = F, v₂ = T). In contrast, {{V1, V2}, {V1, V2}} has exactly two solutions. Show that Double-SAT is NP-hard.

(a) Stingy SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer k, find a satisfying assignment in which at most k variables are true, if such an assignment exists. Prove that stingy SAT is NP-hard. (b) The Double SAT problem asks whether a given satisfiability problem has at least two different satisfying assignments. For example, the problem {{V1, V2}, {V1, V2}, {V1, V2}} is satisfiable, but has only one solution (v₁ = F, v₂ = T). In contrast, {{V1, V2}, {V1, V2}} has exactly two solutions. Show that Double-SAT is NP-hard.