TAOCP 7.2.2.2 Exercise 10

The statement of Exercise 7.

taocpmathematicsalgorithmsvolume-4medium

Section 7.2.2.2: Satisfiability

Exercise 10. ▶ [21] [21] Show that every clause and consistent subset of clauses can be assigned distinct numbers ${1, 2, \ldots, n}$ to the vertices in such a way that each variable, $X_v$, occurs in the clause, or it has size $0$.

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Solve time: 1m53s

Solution

The statement of Exercise 7.2.2.2.10 is incomplete as given. The final condition

$\text{“each variable, }X_v,\text{ occurs in the clause—or it has size }0\text{”}$

does not define a mathematical property, because “the clause,” “the size,” and the role of the vertices and variables $X_v$ are not specified in the supplied Section 7.2.2.2 excerpt.

In particular, the following information is missing:

  1. The object whose vertices are being assigned the numbers ${1,2,\ldots,n}$.
  2. The definition of the variables $X_v$.
  3. The meaning of “a consistent subset of clauses.”
  4. The meaning of “it has size $0$” in the final condition.

Without these definitions, there is no well-defined proposition to prove. Different interpretations lead to different statements, with different proofs and even different truth values.

Please provide the complete text of Exercise 7.2.2.2.10, including the definitions immediately preceding it if they introduce $X_v$ or the relevant graph structure.