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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
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Logistics …Any takers?
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Logistics …Any takers? NO
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Logistics …Any takers? NO
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Logistics …Note: Should now have laptop with you and ready to go with Slate installed.
Any takers? NO
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Logistics …Note: Should now have laptop with you and ready to go with Slate installed.
Any takers? NO
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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
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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
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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
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Your code for Slate & HyperGrader for the semester:
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Your code for Slate & HyperGrader for the semester:
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Your code for Slate & HyperGrader for the semester:
To submit purportedly correct proofs to HyperGrader, you will authenticate with your email address & your code.
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Save sleeve & CD, snapshot sleeve & archive!!
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Save sleeve & CD, snapshot sleeve & archive!!
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What’s on the CD?
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What’s on the CD?
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What’s on the CD?
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What’s on the CD?
LAMA-BDLA
Slate
HyperGrader
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What’s on the CD?
Mac OS
LAMA-BDLA
Slate
HyperGrader
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What’s on the CD?
Mac OSWindows
LAMA-BDLA
Slate
HyperGrader
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What’s on the CD?
Mac OSWindows
textbook
LAMA-BDLA
Slate
HyperGrader
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What’s on the CD?
Mac OSWindows
textbook
LAMA-BDLA
Slate
HyperGrader
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What’s on the CD?
Mac OSWindows
textbook
Complete, sign, email pdf to [email protected]
LAMA-BDLA
Slate
HyperGrader
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Again: Initial Steps
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Again: Initial Steps• Snapshot sleeve with code, and archive.
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Again: Initial Steps• Snapshot sleeve with code, and archive.
• Copy the folder from CD to your laptop.
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Again: Initial Steps• Snapshot sleeve with code, and archive.
• Copy the folder from CD to your laptop.
• Eject CD and “bank-vault”-save both!
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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.
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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
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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.
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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
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–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.”
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A Greek-shocking Example …
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p
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What are rational numbers?p
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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
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
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Page 40
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 43
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
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 47
What are prime numbers?
Page 48
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
And recall: Euclidean “Magic”
Page 50
And recall: Euclidean “Magic”
Page 51
And recall: Euclidean “Magic”
Page 52
And recall: Euclidean “Magic”
Page 53
And recall: Euclidean “Magic”
Page 54
And recall: Euclidean “Magic”
Study it word by work until you endorse it with your very soul!
Page 55
From Algebra 2 in High School …(Pearson Common-Core Compliant Textbook)
Page 58
@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 60
Explosion
PC ⊢ ✓
GIVEN. φ ∧ ¬φ{GIVEN} Assume ✓
GOAL. ψ{GIVEN}
Page 61
Explosion: Partial Proof Plan
Page 62
GreenCheeseMoon1
PC ⊢ ✓
GOAL. G{GIVEN}
GIVEN. ¬(P → (Q → P)){GIVEN} Assume ✓
Page 63
GreenCheeseMoon1: Partial Proof Plan
Sub-Proof Here