DEFINTION2 PropOSItIonal Equvalence 命题 The propositions p and g are called logically equivalent if p t,q is a tautotogy. The notation p e q denotes that p and g are logically equivalent. 逻辑等值,或逻辑等价 2/24/202111:14PM Deren Chen Zhejiang univ
Propositional Equivalences 命题演算 2/24/2021 11:14 PM Deren Chen, Zhejiang Univ. 6 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. ⎯→ 逻辑等值，或逻辑等价
EXAMPLE2 PropOSItIonal Equvalence 命题演算 Show that(pVg) and- pA g 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 (pVg and pA q agree for all possible combinations of the truth values of p and g, it follows that these propositions are logically equivalent. 2/24/202111:14PM Deren Chen, Zhejiang Univ
Propositional Equivalences 命题演算 2/24/2021 11:14 PM Deren Chen, Zhejiang Univ. 7 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.
Table 2 PropOSItIonal Equvalence 命题演算 TABLE 2 Truth Tables for - (pVg and pAq pqp√q|-@p∨q) p PAq T T T T F T F F T T FF F T 2/24/202111:14PM Deren Chen Zhejiang univ
Propositional Equivalences 命题演算 2/24/2021 11:14 PM Deren Chen, Zhejiang Univ. 8 Table 2
EXAMPLE 3 PropOSItIonal Equvalence 命题演 Show that the propositions p-q and -pvgare logically equivalent. Solution: We construct the truth table for these propositions in Table 3. Since the truth values of mpVq and pq agree, these propositions are logically equivalent 2/24/202111:14PM Deren Chen Zhejiang univ
Propositional Equivalences 命题演算 2/24/2021 11:14 PM Deren Chen, Zhejiang Univ. 9 EXAMPLE 3 Show that the propositions p→q and p∨q are logically equivalent. Solution: We construct the truth table for these propositions in Table 3. Since the truth values of p∨q and p→q agree, these propositions are logically equivalent.
Table 3 PropOSItIonal Equvalence 命题演算 TABLE 3 Truth Tables for pvq and p-q p q pVq P→q TTFF TFTF LLⅡd TFTT TFTT 2/24/202111:14PM Deren Chen Zhejiang univ
Propositional Equivalences 命题演算 2/24/2021 11:14 PM Deren Chen, Zhejiang Univ. 10 Table 3