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1 An Introduction to Logic |
be incorporated in the subsequent eliminations. Full details of the method, when applied to integer programming models, are referenced in Sect. 2.6.
1.6 References and Further Work
A good introduction to logic is Mendelson [81]. Another text is Shoenfield [101] . Also Langer [72] is a very clear text. Russell and Whitehead [96] give the results of their formalisation. Godel¨ [44] presents his major results. A ‘popular’ description of Godel’s¨ work is Nagel and Newman [85].
The propositional calculus (Boolean Algebra) is due to Boole [18]. Truth tables were invented by Wittgenstein [124] and Post [90] who also investigated complete connectives. The Sheffer stroke is due to Sheffer [99] and the connective arrow due to Peirce [89].
The compact way of converting statements from DNF to CNF is due to Tseitin [107]. Wilson [123] also presents the method.
The calculus of indications is due to Spencer-Brown [103]. The predicate calculus is usually attributed to Frege [39].
The decision procedure for the theory of dense linear order is due to Langford [73] and that for arithmetic without multiplication is due to Presburger [91].
1.7 Exercises
1.7.1 Use a truth table to show that the following statement is a tautology:
((A B) · (A −→ C) · (B −→ C)) −→ C
1.7.2Verify, by means of truth tables the equivalences (1.4) – (1.21).
1.7.3Check that (1.30) represents the statement in Table 1.2.
1.7.4Use De Morgan’s laws to convert the following statement to DNF:
(A B C) · (B C D) · (A B D)
1.7.5Convert the statement in 1.7.4 into DNF by means of the distributive laws and verify, by means of truth tables, that the statements produced in 1.7.4 and 1.7.5 are equivalent.
1.7.6Convert the following statement into DNF by introducing new variables to represent the disjunctive pairs of statements:
1.7 Exercises |
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(X1 X2) · (X3 X4) · (X5 X6) · · · (X2n−1 X2n )
1.7.7Verify, by means of truth tables, (1.38) – (1.43).
1.7.8Show that ‘|’ and ‘↓’ are the only possible complete connectives of two vari-
ables.
1.7.9Simplify the following expressions:
i.
ii.
iii.
1.7.10Show that applied to any number of statements is a complete connective.
1.7.11Represent the following statements using only the connective :
i.(A B C) → (D E)
ii.(A −→ (B C)) ↔ (B · A)
iii.(A · B · C) B · A)
1.7.12Represent the statement 1.7.11 (i) using a Venn diagram.
1.7.13Represent the following statement using the predicate calculus:
Some animals are mammals but not all animals are mammals although all mammals are animals.
1.7.14 Represent the following statement by the predicate calculus:
24 1 An Introduction to Logic
Ever y even number greater than 2 is ex pressible as the sum o f two prime numbers.
Even numbers and prime numbers should be expressed in terms of more elementary predicates.
1.7.15 Is the following statement true?
x y z(x + y <= z · 2x − 3y z · x > z)
if (i) x, y R (ii) x, y Z.
1.7.16 Show that (1.83) of Example 1.9 is true if the variables are real (or rational).