Download presentation

Presentation is loading. Please wait.

Published byAdam Capper Modified over 2 years ago

1
1 Discrete Structures Lecture 12 Implication III Read: Ch 4.1

2
2 Additional Theorems re: (4.1) p (q p) (4.2) Monotonicity of V: proof in text (p q) (p V r q V r) (4.3) Monotonicity of : proof in class (p q) (p r q r)

3
3 Abbreviation for Proving The transitivity of allows us to use an abbreviation for proofs involving implication. (3.82) Transitivity: (a) (p q) (q r) (p r) P Q R By transitivity (3.82 a), this proves that P R

4
4 Combining and The mutual transitivity of and allows us to use a combination of Leibniz steps and the abbreviation using . (3.82) Transitivity: (b) (p q) (q r) (p r) P Q R By transitivity (3.82b), this proves that P R.

5
5 Combining and The mutual transitivity of and allows us to use a combination of Leibniz steps and the abbreviation using . (3.82) Transitivity: (c) (p q) (q r) (p r) P Q R By transitivity (3.82c), this proves that P R.

6
6 Using the Correct Hints When using this abbreviation, one must be careful to use the correct hints. Example of an incorrect hint: Assuming P1 Q1 is a theorem, we have x P1 x Q1 The correct hint SHOULD be: x P1 x Q1 Incorrect!

7
7 Abbreviation for Proving cont. However, before this can be used, R must first be proved. Here is one proof of R (assuming P1 Q1 is a theorem). (P1 Q1) (x P1 x Q1) = true (x P1 x Q1) = x P1 x Q1

8
8 Abbreviation for Proving cont. This is tortuous, so we can abbreviate as follows. x P1

9
9 (4.3) Monotonicity of : (p q) (p r q r) Problem 4.2 says to prove (4.3) p r q r = ¬(p r) V (q r) = ¬p V ¬r V (q r) = (¬p V ¬r V q) (¬p V ¬r V r) = ¬p V ¬r V q ¬p V q = p q

10
10 A Diversion on Abbreviation Mistakes There may be a tendency to try to write E[z:= P1] E[z:= Q1]

11
11 Diversion Continued The step on the previous slide is not always sound. Here is an unsound use of this technique. ¬P1 ¬Q1 Here is a sound use. ¬P1 ¬Q1

12
12 Metatheorem Parity This metatheorem explains exactly how a replacement of P by Q is to be made, where P Q.

13
13 Metatheorem Parity Continued Metatheorem Parity. Consider a boolean expression E that: contains only operators ¬, V, and and has a single occurrence of a variable z. Suppose P Q is a theorem.

14
14 Metatheorem Parity If z occurs within the scope of an even number of ¬ operations, then E[z:= P] E[z:= Q]. If z occurs within the scope of an odd number of ¬ operations, then E[z:= P] E[z:= Q].

15
15 Metatheorem Parity For example, given P Q, Metatheorem Parity implies the following. R P R Q R ¬P R ¬Q

16
16 Metatheorem Parity For example, given P Q, Metatheorem Parity implies the following. ¬(R P) ¬(R Q) ¬(R ¬P) ¬(R ¬Q)

17
17 Metatheorem Parity For example, given P Q, Metatheorem Parity implies the following. ¬(R P) ¬(R Q) ¬R V ¬P ¬R V ¬Q ¬(R ¬P) ¬(R ¬Q) ¬R V ¬¬P ¬R V ¬¬Q Still even for Q’s scope. In P/Q’s scope, only one negative.

18
18 Bonus Problem (4.4 in book) (p q) (r s) (p V r q V s) Problem 4.4 says to prove the above theorem (p q) (r s)

Similar presentations

Presentation is loading. Please wait....

OK

Introduction to Theorem Proving

Introduction to Theorem Proving

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on tata group of companies Ppt on causes of 1857 revolt leaders Convert video to ppt online free Ppt on agriculture and food security Ppt on machine translation wikipedia Ppt on basic geometrical ideas Ppt on project monitoring and evaluation Ppt on bionics definition Download ppt on indus valley civilization political View my ppt online maker