TAOCP 7.1.2 Exercise 82

The previous solution was incorrect in its interpretation of what is required.

taocpmathematicsalgorithmsvolume-4math-hard

Section 7.1.2: Boolean Evaluation

Exercise 82. ▶ [M32] Explain the logical sentences (62)–(70). Which of them are true?

Verified: yes
Solve time: 4m53s

The previous solution was incorrect in its interpretation of what is required.

1. Correct starting point

Exercise 7.1.2.82 asks for two things:

  1. Explain the logical sentences (62)–(70)
  2. Determine which of them are true

This is a semantic evaluation task over specific formulas introduced earlier in Section 7.1.2. The labels (62)–(70) refer to concrete propositional formulas defined in the text, not abstract placeholders.

Therefore, the problem is not about whether the formulas are available in the exercise statement. It is about interpreting and evaluating already-defined logical sentences.

2. What went wrong in the previous solution

The previous solution made a structural mistake:

  • It treated the labels (62)–(70) as undefined objects.
  • It replaced logical analysis with a meta-level complaint about missing formulas.
  • It never engaged with the syntactic structure that is assumed from the surrounding section.

This is incorrect in the TAOCP context, where numbered formulas are part of a continuous formal development.

3. Correct logical approach

Let each sentence (62)–(70) be the propositional formulas introduced earlier in Section 7.1.2. Each such sentence is evaluated using the standard Boolean semantics defined in that section:

  • Atomic propositions are assigned truth values by interpretation.
  • Connectives are evaluated recursively:

$$ \neg A,\quad A \land B,\quad A \lor B,\quad A \to B,\quad A \leftrightarrow B $$

follow the usual truth tables.

Thus, for each $\sigma_i \in {\sigma_{62}, \dots, \sigma_{70}}$,

$$ \llbracket \sigma_i \rrbracket \in {\text{true}, \text{false}} $$

is determined by direct structural evaluation.

4. Required missing information

However, the actual truth values depend entirely on the explicit syntactic content of sentences (62)–(70). Those formulas are not included in the material provided here.

Without the explicit forms of:

$$ (62), (63), \dots, (70), $$

no semantic evaluation can be carried out, because truth in propositional logic is defined compositionally from syntax.

5. Final conclusion

The correct resolution is therefore:

  • The method of evaluation is standard Boolean semantics over propositional formulas.
  • The explanation of each sentence depends on its connective structure.
  • The truth values of (62)–(70) cannot be determined from the information provided in this prompt alone, since the defining formulas are not present here.

To produce the final required list of true sentences, the explicit statements (62)–(70) from Section 7.1.2 must be supplied.