ln2-page8 - E.g on this last one at line 11 we could have...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
p p 6 6 I The basic structure of an indirect proof is to assume the opposite of what we’re trying to prove, and derive a contradiction. I.e., to show p , we assume p , and show 6 . Let us take as our example: 1. 2. 3. A ! B A ! ⇠ B SHOW: A Pr Pr We set this up by assuming P and adding a new SHOW: line. 1. 2. 3. 4. 5. A ! B A ! ⇠ B SHOW: A A SHOW: 6 Pr Pr ID Ass DD We can show 6 by showing any contradiction. (Any contradiction will do. It need not be the opposite of what we assumed.) 1. 2. 3. 4. 5. 6. 7. 8. A ! B A ! ⇠ B SHOW: A A SHOW: 6 B B 6 Pr Pr ID Ass DD 1,4 ! O 2,4 ! O 6,7 6 I Having shown that a contradiction results when we assume A is true, we can safely conclude that A is false, by ID. 1. 2. 3. 4. 5. 6. 7. 8. A ! B A ! ⇠ B SHOW: A A SHOW: 6 B B 6 Pr Pr ID Ass DD 1,4 ! O 2,4 ! O 6,7 6 I Indirect Derivation is very powerful. It can be used for almost any type of problem, and even when what we’re trying to prove is complex. Here are some additional examples. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. ( P $ Q ) _ ⇠ A A ! P Q SHOW: A A SHOW: 6 P ⇠⇠ A P $ Q P ! Q P 6 Pr Pr Pr ID Ass DD 2,5 ! O 5 DN 1,8 _ O 9 $ O 3,10 ! O 7,11 6 I (There are quite often multiple ways of doing these problems.
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: E.g., on this last one, at line 11, we could have used 7 and 10 to get Q instead of 3 and 10 to get ⇠ P . This would also have given us a different contradiction (lines 3 and 11). Either way you do it is Fne.) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. P ! ( N & M ) Q _⇠ N ⇠ ⇠ ⇠ SHOW: ⇠ ( P & ⇠ Q ) P & ⇠ Q ⇠ ⇠ ⇠ SHOW: 6 P ⇠ Q N & M ⇠ N N 6 Pr Pr ID Ass DD 4 &O 4 &O 1,6 ! O 2,7 _ O 8 &O 9,10 6 I There are, strictly speaking, two forms of indirect proof. The above are all examples of assuming that something is true in order to prove that it is false. We can also assume that something is false in order to prove that it is true. This form works almost exactly the same. Here are two examples. 1. 2. 3. 4. 5. 6. 7. 8. P _ Q Q ! P ⇠ ⇠ ⇠ SHOW: P ⇠ P ⇠ ⇠ ⇠ SHOW: 6 Q P 6 Pr Pr ID Ass DD 1,4 _ O 2,6 ! O 4,7 6 I 8...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern