Condition/Truth Table

Hello, melwin!

Quote:

Are the 2 conditions equivalent? Explain why.

$\text{(a)}\;\text{ Not }(a \leq b)$

. . $(a \geq b)\text{ or Not } (a=b)$

$a \leq b$ means: $(a$ is less than $b)$ or ( $a$ equal to $b.)$

Its negation is: . $\sim(a \leq b) \quad=\quad\sim\bigg[(a\text{ is less than }b) \text{ or } (a = b)\bigg]$

By DeMorgan's Law, this is: . $\sim(a\text{ is less than }b)\:\text{ and }\sim(a - b)$

. . . . . . . . . . . . . . . . $=\;(a\text{ is greater than or equal to }b)\;\text{ and }\:(a \neq b)$

. . . . . . . . . . . . . . . . . . . . $=\;(a \geq b) \text{ {\color{red}and} Not }(a = b)$

This is not what the second statement says; they are not equivalent.

Quote:

$\text{(b)}\;\text{Not }(a \leq b)$

. . $(a \geq b)\text{ and Not }(a = b)$

These two are equivalent . . . see part (a).

Quote:

Write a statement equivalent to the negation of the given condition
that does not use the NOT operator.

$\text{(a)}\;a < b$

$\sim(a < b) \quad\Longrightarrow\quad a \geq b$

Quote:

$\text{(b) }\;(a>b) \wedge (c \neq d)$

$\sim\bigg[(a > b) \wedge (c \neq d)\bigg] \quad\Rightarrow\quad \sim(a > b)\: \vee \sim(c \neq d) \quad\Rightarrow\quad (a \leq b) \vee (c = d)<br />$

Quote:

$\text{(c) }\;(a=b) \vee(a=c)$

$\sim\bigg[(a=b) \vee (a = c)\bigg] \quad\Rightarrow\quad \sim(a=b)\: \wedge \sim(a = c) \quad\Rightarrow\quad (a \neq b) \:\wedge (a \neq c)<br /> <br />$