《命题演算》PPT课件

逻辑等值，或逻辑等价
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Deren Chen, Zhejiang Univ.
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EXAMPLE 2
• Show that (p∨q) and p∧ q are logically equivalent. This equivalence is one
of De Morgan's laws for propositions, named after the English mathematician
Augustus De Morgan, of the mid-nineteenth century.
Solution: The truth tables for these propositions are displayed in Table 2. Since the truth values of the propositions (p∨q) and p∧ q agree for all possible combinations of the truth values of p and q, it follows that these propositions are logically equivalent.
Solution: We construct the truth table for these propositions in Table 3. Since
the truth values of equivalent.
p∨q and p→q agree, these propositions are logically
Since p∨ p is always true, it is a tautology. Since p∧ p is always false, it is a
contradicthen, Zhejiang Univ.
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Table 1
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• 永真命题公式（Tautology） • 公式中的命题变量无论怎样代入，公式对应的真值恒为T。 • 永假命题公式（Contradiction) • 公式中的命题变量无论怎样代入，公式对应的真值恒为F。 • 可满足命题公式（Satisfaction） • 公式中的命题变量无论怎样代入，公式对应的真值总有一种情况为T。 • 一般命题公式（Contingency） • 既不是永真公式也不是永假公式。
11/21/2020 5:28 AM
Deren Chen, Zhejiang Univ.
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Table 2
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Deren Chen, Zhejiang Univ.
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EXAMPLE 3
• Show that the propositions p→q and p∨q are logically equivalent.
Deren Chen, Zhejiang Univ.
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DEFINITION 2
• The propositions p and q are called logically equivalent if p q is a tautotogy. The notation p q denotes that p and q are logically equivalent.
• 1.2 命题演算 • Propositional Equivalences
11/21/2020 5:28 AM
Deren Chen, Zhejiang Univ.
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• 1、命题(Proposition)
• 2、从简单命题(atomic proposition)到
•
复合命题(compositional proposition)
equivalent.This is the distributive law of disjunction over conjunction.
Solution: We construct the truth table for these propositions in Table 4. Since the truth values of p∨(q∧r) and (p∨q)∧(p∨r) agree, these propositions are logically equivalent.
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Deren Chen, Zhejiang Univ.
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Table 4
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Deren Chen, Zhejiang Univ.
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• 基本逻辑等价定理：
• 对于任意的命题公式p、q、r，下面的命题公 式是等价的。
11/21/2020 5:28 AM
Deren Chen, Zhejiang Univ.
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EXAMPLE 1
• We can construct examples of tautologies and contradictions using just one
proposition. Consider the truth tables of p∨ p and p∧ p, shown in Table 1.
• 3、从命题常量(propositional constant)到
•
命题变量(propositional variable)
• 4、从复合命题(compositional proposition)到
•
命题公式(propositional formulas)
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Deren Chen, Zhejiang Univ.
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Deren Chen, Zhejiang Univ.
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Table 3
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EXAMPLE 4
• Show that the propositions p∨(q∧r) and (p∨q)∧(p∨r) are logically