chapter 8

Section 8.1

1.  1.       {odd(x + 1)} y : = x + 1 {odd(y)}        AA

2.        True ∧ even(x )        P [for CP]

3.        even(x )          2, Simp

4.        odd(x + 1)        3, T

5.    True ∧ even(x ) → odd(x + 1)           2−4, CP

6. {True ∧ even(x ) } y := x + 1 {odd(y)}     1, 5, Consequence

QED.

2. a. 1.   $\left\{x+\text{\hspace{0.17em}}b>\text{\hspace{0.17em}}0\right\}\text{\hspace{0.17em}}y:\text{\hspace{0.17em}}=\text{\hspace{0.17em}}b\left\{x+y>0\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}AA$

2.   $\left\{a+\text{\hspace{0.17em}}b>\text{\hspace{0.17em}}0\right\}\text{\hspace{0.17em}}x:\text{\hspace{0.17em}}=\text{\hspace{0.17em}}a\left\{x+b>0\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}AA$

3.    $\left(a>0\right)\wedge \left(b>\text{\hspace{0.17em}}0\right)\to \text{\hspace{0.17em}}\left(a+\text{\hspace{0.17em}}b>0\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}T$

4.    $\left\{\left(a>\text{\hspace{0.17em}}0\right)\text{\hspace{0.17em}}\wedge \text{\hspace{0.17em}}\left(b>\text{\hspace{0.17em}}0\right)$

Get Discrete Structures, Logic, and Computability, 4th Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.