Section 2.2

3. (a) $q\rightarrow p$

(b) MATH

(c) MATH

5. Note that if $p\rightarrow q$ is known to be false, then it must be the case that $p=1$ and $q=0$ (see Truth Table in middle of page 81 of text.) Therefore,

(a)

$p$ $q$ $p\wedge q$
$1$ $0$ $0$
So the truth value is $0$.

(b)

$p$ $q$ $p\vee q$
$1$ $0$ $1$
So the truth value is $1$.

(c)

$p$ $q$ $q\rightarrow p$
$1$ $0$ $1$
So the truth value is $1$.

8.

$p$ $q$ MATH $\rightarrow $ MATH $\rightarrow $ MATH
$0$ $0$ $1$ $0$ $1$ $0$ $0$
$0$ $1$ $1$ $1$ $0$ $1$ $1$
$1$ $0$ $0$ $1$ $1$ $1$ $1$
$1$ $1$ $1$ $1$ $1$ $1$ $1$
(Note: the ''*'' marks the column of interest.)

*

10.

$p$ $q$ $r$ MATH $\vee $ MATH $\rightarrow $ MATH
$0$ $0$ $0$ $1$ $1$ $1$ $0$ $0$
$0$ $0$ $1$ $1$ $1$ $1$ $0$ $0$
$0$ $1$ $0$ $0$ $1$ $1$ $0$ $0$
$0$ $1$ $1$ $0$ $1$ $1$ $0$ $0$
$1$ $0$ $0$ $0$ $0$ $0$ $1$ $1$
$1$ $0$ $1$ $0$ $1$ $1$ $1$ $1$
$1$ $1$ $0$ $1$ $1$ $0$ $0$ $0$
$1$ $1$ $1$ $1$ $1$ $1$ $0$ $0$
(Note: the ''*'' marks the column of interest.)

*

11. (a)

$p$ $q$ $r$ $\lnot $ MATH $\rightarrow $ $r$
$0$ $0$ $0$ $1$ $0$ $0$ $0$
$0$ $0$ $1$ $1$ $0$ $1$ $1$
$0$ $1$ $0$ $0$ $1$ $1$ $0$
$0$ $1$ $1$ $0$ $1$ $1$ $1$
$1$ $0$ $0$ $0$ $1$ $1$ $0$
$1$ $0$ $1$ $0$ $1$ $1$ $1$
$1$ $1$ $0$ $0$ $1$ $1$ $0$
$1$ $1$ $1$ $0$ $1$ $1$ $1$
(b)
$p$ $q$ $r$ $\lnot $ MATH $\rightarrow $ $r) $
$0$ $0$ $0$ $0$ $0$ $1$ $0$
$0$ $0$ $1$ $0$ $0$ $1$ $1$
$0$ $1$ $0$ $1$ $1$ $0$ $0$
$0$ $1$ $1$ $0$ $1$ $1$ $1$
$1$ $0$ $0$ $1$ $1$ $0$ $0$
$1$ $0$ $1$ $0$ $1$ $1$ $1$
$1$ $1$ $0$ $1$ $1$ $0$ $0$
$1$ $1$ $1$ $0$ $1$ $1$ $1$

* *

(Note: the ''*'' marks the columns of interest.)

18. (c)

$p$ $q$ $r$ MATH $\rightarrow $ $r\rbrack $ MATH $\lbrack p$ $\rightarrow $ MATH
$0$ $0$ $0$ $1$ $0$ $0$ $0$ $0$ $1$ $1$
$0$ $0$ $1$ $1$ $1$ $1$ $1$ $0$ $1$ $1$
$0$ $1$ $0$ $1$ $0$ $0$ $0$ $0$ $1$ $0$
$0$ $1$ $1$ $1$ $1$ $1$ $1$ $0$ $1$ $1$
$1$ $0$ $0$ $0$ $1$ $0$ $1$ $1$ $1$ $1$
$1$ $0$ $1$ $0$ $1$ $1$ $1$ $1$ $1$ $1$
$1$ $1$ $0$ $1$ $0$ $0$ $1$ $1$ $0$ $0$
$1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$

*

So this statement is false, since the column of interest (marked ''*'') is not all $1$'s.

(d)

$p$ $q$ $r$ MATH $\leftrightarrow $ $r\rbrack $ MATH $\lbrack p$ $\leftrightarrow $ MATH
$0$ $0$ $0$ $1$ $0$ $0$ $1$ $0$ $0$ $1$
$0$ $0$ $1$ $1$ $1$ $1$ $1$ $0$ $1$ $0$
$0$ $1$ $0$ $0$ $1$ $0$ $1$ $0$ $1$ $0$
$0$ $1$ $1$ $0$ $0$ $1$ $1$ $0$ $0$ $1$
$1$ $0$ $0$ $0$ $1$ $0$ $1$ $1$ $1$ $1$
$1$ $0$ $1$ $0$ $0$ $1$ $1$ $1$ $0$ $0$
$1$ $1$ $0$ $1$ $0$ $0$ $1$ $1$ $0$ $0$
$1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$

*

So this statement is true, since the column of interest (marked ''*'') is all $1$'s.

20. (a)

$p$ $q$ MATH $\wedge $ $\lnot q\rbrack $ $\Rightarrow $ $\lnot p$
$0$ $0$ $1$ $1$ $1$ $1$ $1$
$0$ $1$ $1$ $0$ $0$ $1$ $1$
$1$ $0$ $0$ $0$ $1$ $1$ $0$
$1$ $1$ $1$ $0$ $0$ $1$ $0$

*

This verifies Rule 20 since in the column of interest (marked ''*''), all of the entries are $1$'s.

(b)

$p$ $q$ MATH $\wedge $ $\lnot p\rbrack $ $\Rightarrow $ $q$
$0$ $0$ $0$ $0$ $1$ $1$ $0$
$0$ $1$ $1$ $1$ $1$ $1$ $1$
$1$ $0$ $1$ $0$ $0$ $1$ $0$
$1$ $1$ $1$ $0$ $0$ $1$ $1$

*

This verifies Rule 21 since in the column of interest (marked ''*''), all of the entries are $1$'s.

This document created by Scientific Notebook 4.1.