Logical symbols [Mathematical Appendix : Logic]
・connectives:negation ¬~ / disjunction
∨ /conjunction ∧ / implication,conditional⇒
/ equivalent ⇔ |
negation : ¬P, ~P |
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・ "¬P" "~P" means "not
P"。 ・The truth-table below defines the truth value of"¬P".
* Whether "¬P" is true or false [in detail] →Semantics "¬P " |
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・"P∨Q means "P or
Q" . ・The truth-table below defines the truth value of "P∨Q".
* Whether "P∨Q" is true or false |
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* The two expressions below are interchangeable. ・ ¬ ( A ∨ B ) ・ ( ¬ A ) ∧ ( ¬ B ) |
[reference] ・Sugiura , Introduction to Analysis I , 399; ・Motohashi『新しい論理序 説(New Introduction to Logic)』6.4問題6(p.121;128) ・Iritani and Kuga , Introduction to Mathematical Economics,定理1.1(p.4) |
→[logical symbols :contents] →[Mathematical Appendix : Logic] →[Mathematical Appendixes] |
・"P∧Q means "P and
Q" "P but Q". ・The truth-table below defines the truth value of "P∧Q".
* Whether "P∧Q" is true or false |
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* The two expressions below are interchangeable. ・ ¬ ( A ∧ B ) ・ ( ¬ A ) ∨ ( ¬ B ) |
[reference] ・Sugiura , Introduction to Analysis I , 399; ・Motohashi『新しい論理序 説(New Introduction to Logic)』6.4問題6(p.121;128) ・Iritani and Kuga , Introduction to Mathematical Economics,定理1.1(p.4) |
→[logical symbols :contents] →[Mathematical Appendix : Logic] →[Mathematical Appendixes] |
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・"P⇔Q" means "P if and only if (iff) Q". |
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・The truth-table below defines the truth value of "P⇔Q".
* The two expressions below are interchangeable. ・ P⇔Q ・ (P⇒Q)∧(P⇒Q) |
→[logical symbols :contents] →[Mathematical Appendix : Logic] →[Mathematical Appendixes] |
・"P⇒Q" means "if P , thenQ" "Q,if P ","P,only if Q" ,"when P , Q"。 [→詳細] ・The truth-table below defines the truth value of "A⇒B".
[→詳細] |
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* The two expressions below are always interchangeable. ・ ¬( A ⇒ B ) ・ A ∧ ¬ ( B ) |
[reference] ・Sugiura , Introduction to Analysis I , 400; ・Iritani and Kuga『数 理経済学入門』定理1.1(7)(p.4) |
* The two expressions below are always interchangeable. ∀x ( P(x) ∧ Q(x) ) ( ∀x P(x) ) ∧ ( ∀x Q(x) ) * The two expressions below are always interchangeable. ∀x∈S ( P(x) ∧ Q(x) ) ( ∀x∈S P(x) ) ∧ ( ∀x∈S Q(x) ) |
[reference] ・Nakauchi, Logic Workbook,定理2.2.10(p.87):∀(∨) is also mentioned. ・Nakatani, Logic, 4.3-C-(4・5)(p.90):∀(∨) is also mentioned. ・Saitoh , from Japanese to Symbolic Logic, 2 章§7付録(pp.94-95) |
* The two expressions below are always interchangeable. ∀x ∀y P(x,y) ∀y ∀x P(x,y) * The two expressions below are always interchangeable. ∀x∈S ∀y∈T P(x,y) ∀y∈T ∀x∈S P(x,y) |
→[logical symbols :contents] →[Mathematical Appendix : Logic] →[Mathematical Appendixes] |
* The two expressions below are interchangeable. ∃x ( P(x) ∨ Q(x) ) ( ∃x P(x) ) ∨ ( ∃x Q(x) ) * The two expressions below are interchangeable. ∃x∈S ( P(x) ∨ Q(x) ) ( ∃x∈S P(x) ) ∨ ( ∃x∈S Q(x) ) |
[reference] ・Nakauchi, Logic Workbook,定理2.2.10(p.87):∃(∧) is also mentioned. ・Nakatani, Logic, 4.3-C-(4・5)(p.90):∃(∧)is also mentioned. |
* The two expressions below are always interchangeable. ∃x ∃y P(x,y) ∃y ∃x P(x,y) * The two expressions below are always interchangeable. ∃x∈S ∃y∈T P(x,y) ∃y∈T ∃x∈S P(x,y) |
→[logical symbols :contents] →[Mathematical Appendix : Logic] →[Mathematical Appendixes] |
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partial negation - two expressions |
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【1】 The two expressions below are always interchangeable. ・¬ ( ∀x P(x) ) ・∃x (¬P(x) ) 【2】 The two expressions below are always interchangeable. ・¬ ( ∀x∈S P(x) ) ・∃x∈S (¬P(x) ) |
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total negation - two expressions
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【1】 The two expressions below are always interchangeable. ¬ ( ∃x P(x) ) ∀x (¬ P(x) ) 【2】 The two expressions below are always interchangeable. ¬ ( ∃x∈S P(x) ) ∀x∈S (¬ P(x) ) |
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→[logical symbols :contents] →[Mathematical Appendixes : Logic] →[Mathematical Appendixes] |
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→[logical symbols :contents] →[Mathematical Appendix : Logic] →[Mathematical Appendixes] |