as propositional calculus iii - rensselaer polytechnic...
TRANSCRIPT
![Page 1: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/1.jpg)
Propositional Calculus III: Reductio ad Absurdum
Rensselaer AI & Reasoning (RAIR) LabDepartment of Cognitive ScienceDepartment of Computer Science
Lally School of Management & TechnologyRensselaer Polytechnic Institute (RPI)
Troy, New York 12180 USA
Intro to Logic2/8/2018; & again 2/12/18 by Rini Palamittam
Selmer Bringsjord
![Page 2: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/2.jpg)
Logistics …
![Page 3: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/3.jpg)
Logistics …Any takers?
![Page 4: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/4.jpg)
Logistics …Any takers? NO
![Page 5: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/5.jpg)
Logistics …Any takers? NO
![Page 6: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/6.jpg)
Logistics …Note: Should now have laptop with you and ready to go with Slate installed.
Any takers? NO
![Page 7: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/7.jpg)
Logistics …Note: Should now have laptop with you and ready to go with Slate installed.
Any takers? NO
![Page 8: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/8.jpg)
Logistics …Note: Should now have laptop with you and ready to go with Slate installed.
And … HyperGrader will debut in class on Feb 12, led by Rini.
Any takers? NO
![Page 9: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/9.jpg)
Logistics …Note: Should now have laptop with you and ready to go with Slate installed.
And … HyperGrader will debut in class on Feb 12, led by Rini.
http://www.logicamodernapproach.com
Any takers? NO
![Page 10: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/10.jpg)
Logistics …Note: Should now have laptop with you and ready to go with Slate installed.
And … HyperGrader will debut in class on Feb 12, led by Rini.
http://www.logicamodernapproach.com
Plenty of problems there to already learn/practice on!
Any takers? NO
![Page 11: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/11.jpg)
Your code for Slate & HyperGrader for the semester:
![Page 12: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/12.jpg)
Your code for Slate & HyperGrader for the semester:
![Page 13: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/13.jpg)
Your code for Slate & HyperGrader for the semester:
To submit purportedly correct proofs to HyperGrader, you will authenticate with your email address & your code.
![Page 14: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/14.jpg)
Save sleeve & CD, snapshot sleeve & archive!!
![Page 15: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/15.jpg)
Save sleeve & CD, snapshot sleeve & archive!!
![Page 16: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/16.jpg)
What’s on the CD?
![Page 17: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/17.jpg)
What’s on the CD?
![Page 18: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/18.jpg)
What’s on the CD?
![Page 19: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/19.jpg)
What’s on the CD?
LAMA-BDLA
Slate
HyperGrader
![Page 20: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/20.jpg)
What’s on the CD?
Mac OS
LAMA-BDLA
Slate
HyperGrader
![Page 21: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/21.jpg)
What’s on the CD?
Mac OSWindows
LAMA-BDLA
Slate
HyperGrader
![Page 22: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/22.jpg)
What’s on the CD?
Mac OSWindows
textbook
LAMA-BDLA
Slate
HyperGrader
![Page 23: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/23.jpg)
What’s on the CD?
Mac OSWindows
textbook
LAMA-BDLA
Slate
HyperGrader
![Page 24: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/24.jpg)
What’s on the CD?
Mac OSWindows
textbook
Complete, sign, email pdf to [email protected]
LAMA-BDLA
Slate
HyperGrader
![Page 25: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/25.jpg)
Again: Initial Steps
![Page 26: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/26.jpg)
Again: Initial Steps• Snapshot sleeve with code, and archive.
![Page 27: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/27.jpg)
Again: Initial Steps• Snapshot sleeve with code, and archive.
• Copy the folder from CD to your laptop.
![Page 28: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/28.jpg)
Again: Initial Steps• Snapshot sleeve with code, and archive.
• Copy the folder from CD to your laptop.
• Eject CD and “bank-vault”-save both!
![Page 29: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/29.jpg)
Again: Initial Steps• Snapshot sleeve with code, and archive.
• Copy the folder from CD to your laptop.
• Eject CD and “bank-vault”-save both!
• Depending upon whether you’re Windows or MacOS, expand the relevant zipped file to obtain Slate.
![Page 30: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/30.jpg)
Again: Initial Steps• Snapshot sleeve with code, and archive.
• Copy the folder from CD to your laptop.
• Eject CD and “bank-vault”-save both!
• Depending upon whether you’re Windows or MacOS, expand the relevant zipped file to obtain Slate.
• Open Slate
![Page 31: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/31.jpg)
Again: Initial Steps• Snapshot sleeve with code, and archive.
• Copy the folder from CD to your laptop.
• Eject CD and “bank-vault”-save both!
• Depending upon whether you’re Windows or MacOS, expand the relevant zipped file to obtain Slate.
• Open Slate
• Today I’ll continue to explain and show inference rules in Slate.
![Page 32: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/32.jpg)
Propositional Calculus III: Reductio ad Absurdum
Rensselaer AI & Reasoning (RAIR) LabDepartment of Cognitive ScienceDepartment of Computer Science
Lally School of Management & TechnologyRensselaer Polytechnic Institute (RPI)
Troy, New York 12180 USA
Intro to Logic2/8/2018; & again on 2/12/18 by Rini Palamittam
Selmer Bringsjord
![Page 33: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/33.jpg)
Reductio …
![Page 34: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/34.jpg)
–G. H. Hardy
“Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.”
![Page 35: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/35.jpg)
A Greek-shocking Example …
![Page 36: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/36.jpg)
p
q<latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit>
![Page 37: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/37.jpg)
What are rational numbers?p
q<latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit>
![Page 38: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/38.jpg)
What are rational numbers?• Any number that can be expressed in the form of
such that we have a numerator p and a non-zerodenominator.
• Rational numbers are a subset of real numbers!
• Examples of irrational numbers?
p
q<latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit>
![Page 39: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/39.jpg)
What are rational numbers?• Any number that can be expressed in the form of
such that we have a numerator p and a non-zerodenominator.
• Rational numbers are a subset of real numbers!
• Examples of irrational numbers?
p
q<latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit>
p2
<latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit>
![Page 40: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/40.jpg)
What are rational numbers?• Any number that can be expressed in the form of
such that we have a numerator p and a non-zerodenominator.
• Rational numbers are a subset of real numbers!
• Examples of irrational numbers?
p
q<latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit><latexit sha1_base64="wgdOTsBURTntJpkDMU0eoBw650I=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzW7c3Qgl5Gd48aDi1X/jzX/jts1BWx8MPN6bYWZelHKmjet+O5WV1bX1jepmbWt7Z3evvn/woGWmCPWJ5FJ1I6wpZ4L6hhlOu6miOIk47UTjm6nfeaJKMynuzSSlYYKHgsWMYGOloBcrTPK0yB+Lfr3hNt0Z0DLxStKAEu1+/as3kCRLqDCEY60Dz01NmGNlGOG0qPUyTVNMxnhIA0sFTqgO89nJBTqxygDFUtkSBs3U3xM5TrSeJJHtTLAZ6UVvKv7nBZmJL8OciTQzVJD5ojjjyEg0/R8NmKLE8IklmChmb0VkhG0KxqZUsyF4iy8vE/+sedV0784bresyjSocwTGcggcX0IJbaIMPBCQ8wyu8OcZ5cd6dj3lrxSlnDuEPnM8fYyyRhA==</latexit>
p2
<latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit>
e<latexit sha1_base64="8g0P2hkn1tmY+V1wE8v60UKiD0I=">AAAB53icbVBNS8NAEJ34WetX1aOXxSJ4KokI6q3oxWMLxhbaUDbbSbt2swm7G6GE/gIvHlS8+pe8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6CRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTv3WEyrNE3lvxikGMR1IHnFGjZWa2KtU3Zo7A1kmXkGqUKDRq3x1+wnLYpSGCap1x3NTE+RUGc4ETsrdTGNK2YgOsGOppDHqIJ8dOiGnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrRm4r/eZ3MRFdBzmWaGZRsvijKBDEJmX5N+lwhM2JsCWWK21sJG1JFmbHZlG0I3uLLy8Q/r13X3OZFtX5TpFGCYziBM/DgEupwBw3wgQHCM7zCm/PovDjvzse8dcUpZo7gD5zPHzeUjLw=</latexit><latexit sha1_base64="8g0P2hkn1tmY+V1wE8v60UKiD0I=">AAAB53icbVBNS8NAEJ34WetX1aOXxSJ4KokI6q3oxWMLxhbaUDbbSbt2swm7G6GE/gIvHlS8+pe8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6CRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTv3WEyrNE3lvxikGMR1IHnFGjZWa2KtU3Zo7A1kmXkGqUKDRq3x1+wnLYpSGCap1x3NTE+RUGc4ETsrdTGNK2YgOsGOppDHqIJ8dOiGnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrRm4r/eZ3MRFdBzmWaGZRsvijKBDEJmX5N+lwhM2JsCWWK21sJG1JFmbHZlG0I3uLLy8Q/r13X3OZFtX5TpFGCYziBM/DgEupwBw3wgQHCM7zCm/PovDjvzse8dcUpZo7gD5zPHzeUjLw=</latexit><latexit sha1_base64="8g0P2hkn1tmY+V1wE8v60UKiD0I=">AAAB53icbVBNS8NAEJ34WetX1aOXxSJ4KokI6q3oxWMLxhbaUDbbSbt2swm7G6GE/gIvHlS8+pe8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw+6CRTDH2WiES1Q6pRcIm+4UZgO1VI41BgKxzdTv3WEyrNE3lvxikGMR1IHnFGjZWa2KtU3Zo7A1kmXkGqUKDRq3x1+wnLYpSGCap1x3NTE+RUGc4ETsrdTGNK2YgOsGOppDHqIJ8dOiGnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrRm4r/eZ3MRFdBzmWaGZRsvijKBDEJmX5N+lwhM2JsCWWK21sJG1JFmbHZlG0I3uLLy8Q/r13X3OZFtX5TpFGCYziBM/DgEupwBw3wgQHCM7zCm/PovDjvzse8dcUpZo7gD5zPHzeUjLw=</latexit>
![Page 41: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/41.jpg)
![Page 42: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/42.jpg)
Prove that:
![Page 43: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/43.jpg)
Prove that:p2
<latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit>
![Page 44: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/44.jpg)
Prove that:p2
<latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit><latexit sha1_base64="eaQNQENQGoHK9B+awuZhUywg7xY=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoN6KXjxWsLbQhrLZbtqlm026OxFK6J/w4kHFq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHj0aOJUM95ksYx1O6CGS6F4EwVK3k40p1EgeSsY3c781hPXRsTqAScJ9yM6UCIUjKKV2l0z1pjVpr1yxa26c5BV4uWkAjkavfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzyM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0mzVr2uuvcXlfpNnkYRTuAUzsGDS6jDHTSgCQwkPMMrvDlj58V5dz4WrQUnnzmGP3A+fwCvNo/t</latexit>
is irrational
![Page 45: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/45.jpg)
![Page 46: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/46.jpg)
![Page 47: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/47.jpg)
What are prime numbers?
![Page 48: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/48.jpg)
What are prime numbers?
• A number that can be divided by only twonumbers, one and itself.
• Must be a whole number.
• Example: 2,3,5,7……
![Page 49: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/49.jpg)
And recall: Euclidean “Magic”
![Page 50: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/50.jpg)
And recall: Euclidean “Magic”
![Page 51: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/51.jpg)
And recall: Euclidean “Magic”
![Page 52: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/52.jpg)
And recall: Euclidean “Magic”
![Page 53: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/53.jpg)
And recall: Euclidean “Magic”
![Page 54: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/54.jpg)
And recall: Euclidean “Magic”
Study it word by work until you endorse it with your very soul!
![Page 55: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/55.jpg)
From Algebra 2 in High School …(Pearson Common-Core Compliant Textbook)
![Page 56: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/56.jpg)
![Page 57: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/57.jpg)
![Page 58: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/58.jpg)
@article{moore.proof, Author = {R. C. Moore}, Journal = {Educational Studies in Mathematics}, Pages = {249-266}, Title = {Making the Transition to Formal Proof}, Volume = {27.3}, Year = 1994}
![Page 59: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/59.jpg)
![Page 60: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/60.jpg)
Explosion
PC ⊢ ✓
GIVEN. φ ∧ ¬φ{GIVEN} Assume ✓
GOAL. ψ{GIVEN}
![Page 61: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/61.jpg)
Explosion: Partial Proof Plan
![Page 62: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/62.jpg)
GreenCheeseMoon1
PC ⊢ ✓
GOAL. G{GIVEN}
GIVEN. ¬(P → (Q → P)){GIVEN} Assume ✓
![Page 63: AS propositional calculus III - Rensselaer Polytechnic Institutekryten.mm.rpi.edu/.../AS_propositional_calculus_III.pdfPropositional Calculus III: Reductio ad Absurdum Rensselaer AI](https://reader034.vdocuments.us/reader034/viewer/2022042605/5f527de8f933282f2e7f1f1e/html5/thumbnails/63.jpg)
GreenCheeseMoon1: Partial Proof Plan
Sub-Proof Here