ud 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... ·...
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![Page 1: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/1.jpg)
• ����– UD�2�3�4�
![Page 2: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/2.jpg)
��1�logically speaking
• ������logically speaking��–���statement statement form�����– statement form��������–������������
![Page 3: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/3.jpg)
%&1�logically speaking ( )
• ��"$�statement form������#��������– negation����– disjunction����– conjunction����– implication�!��– equivalence���
– tautology�����– contradiction�� ��
![Page 4: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/4.jpg)
��2�truth table
• ������statement form���� �������������
![Page 5: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/5.jpg)
��2�truth table (�)
• �������������plan ���– If it is Wednesday, then Mr. French eats only pickles.– If it is Monday, then Mr. French eats only chocolate.– Mr. French is eating chocolate.– ����� ����
![Page 6: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/6.jpg)
��2�truth table (�)
• �����������������– Mr. Hamburger is German or Swiss.– Mr. Hamburger is not Swiss.–���Mr. Hamburger�� ��
![Page 7: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/7.jpg)
��2�truth table (�)
• �����������������– Mr. Hamburger is German or Swiss.– Mr. Hamburger is not Swiss.–���Mr. Hamburger�� ��
G S G∨S ¬ST T T FT F T TF T T FF F F T
![Page 8: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/8.jpg)
��2�truth table ( )
• �����������������– Knights and Knaves• John: We are both knaves.• Bill: …
![Page 9: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/9.jpg)
��2�truth table ( )
• �����������������– Knights and Knaves• John: We are both knaves.• Bill: …
J B ¬J∧¬B J↔(¬J∧¬B)T T F FT F F FF T F TF F T F
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��2�truth table ( )
• �����������������– Knights and Knaves• John: We are the same kind.• Bill: We are of different kinds.
![Page 11: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/11.jpg)
��2�truth table ( )
• �����������������– Knights and Knaves• John: We are the same kind.• Bill: We are of different kinds.
J B (J↔(J↔B))∧(B↔¬(J↔B))T T FT F FF T TF F F
![Page 12: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/12.jpg)
��3�equivalent statement forms
• ���equivalent statement forms�• ��������equivalence���� �• �����������
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#$3�equivalent statement forms (�)
• � � �����!�����statement form����equivalent statement form�– A∨B– A∧B
• ����"�������
![Page 14: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/14.jpg)
#$3�equivalent statement forms (�)
• � � �����!�����statement form����equivalent statement form�– A∨B�¬A→B– A∧B�¬(A→¬B)
• ����"�������
![Page 15: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/15.jpg)
��3�equivalent statement forms (�)
• ��������������statement form� ��equivalent statement form�– ¬A– A∧B– A∨B
![Page 16: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/16.jpg)
��3�equivalent statement forms (�)
• �������������statement form���equivalent statement form�– ¬A– A∧B– A∨B
![Page 17: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/17.jpg)
��3�equivalent statement forms (�)
• �������������statement form���equivalent statement form�– ¬A– A∧B– A∨B
![Page 18: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/18.jpg)
��3�equivalent statement forms (�)
• ����� ��equivalent statement forms����������
![Page 19: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/19.jpg)
��3�equivalent statement forms (�)
• �������������������� ��paraphrase����– Knights and Knaves• John: We are both knaves.• Bill: …
J B ¬J∧¬B J↔(¬J∧¬B)T T F FT F F FF T F TF F T F
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(J∧(¬J∧¬B))∨(¬J∧¬(¬J∧¬B))= (J∧¬J∧¬B)∨(¬J∧(¬¬J∨¬¬B))=F∨(¬J∧(J∨B))=¬J∧(J∨B)=(¬J∧J)∨(¬J∧B)=F∨(¬J∧B)=¬J∧B
��3�equivalent statement forms (�)
![Page 21: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/21.jpg)
(J∧(¬J∧¬B))∨(¬J∧¬(¬J∧¬B))= (J∧¬J∧¬B)∨(¬J∧(¬¬J∨¬¬B))=F∨(¬J∧(J∨B))=¬J∧(J∨B)=(¬J∧J)∨(¬J∧B)=F∨(¬J∧B)=¬J∧B���������� ������
��3�equivalent statement forms (�)
![Page 22: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/22.jpg)
��4�set notation and quantifiers
• ������• ������� ��������– extensional definition• {-1, 1}• {1}
– intensional definition• {2n : n∈Z}• {(m,n)∈R2 : y=0}
![Page 23: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/23.jpg)
��4�set notation and quantifiers ()
• ������� ��
![Page 24: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/24.jpg)
• �������������– For all x∈A, property p(x) holds.– For some x∈A, property p(x) holds.
��4�set notation and quantifiers ()
![Page 25: UD 2 3 4 - cslabcms.nju.edu.cncslabcms.nju.edu.cn › problem_solving › images › b › ba... · –UD 2 3 4 1 logically speaking • logically speaking – statement statement](https://reader036.vdocuments.net/reader036/viewer/2022062303/5f2524066f0da472197a8649/html5/thumbnails/25.jpg)
• �������������– For all x∈A, property p(x) holds.– For some x∈A, property p(x) holds.
( )( )xpAxx ®Î" ,( )( )xpAxx ÙÎ$ ,
��4�set notation and quantifiers ()
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• ���� ��������– For all x∈A, property p(x) holds.– For some x∈A, property p(x) holds.
• �������� �
( )( )xpAxx ®Î" ,( )( )xpAxx ÙÎ$ ,
( )( )xpAxx ®Î$ ,
��4�set notation and quantifiers (�)
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• ���� �
��4�set notation and quantifiers (�)